The first thoughts about artificial intelligence

Mechanical machines

Robots and Automatic Machines Were Generally Very Inventive: Al-Jazari (XII Century)

Hesdin Castle (Robert II of Artois), Leonardo’s robot…

Mechanical machines

Jaquet-Droz automata (XVIII century):

Mechanical machines

• But this is in mechanics, in mathematics/logic AI it was quite rudimentary for a long time

Logic machine of Ramon Llull (XIII-XIV centuries)

• Starting with Dr. Frankenstein, further AI in the literature appears constantly …

Turing Test

• AI as a science begins with a Turing test (1950).
• The idea of the Turing test is to check if a machine can imitate a human in a conversation.
• The original formulation was more nuanced.

Shannon’s Theseus

• YouTube Video
• Early 1950s, Claude Shannon (The father of Information Theory) demonstrates Theseus
• A life-sized magnetic mouse controlled by relay circuits, learns its way around a maze.

Stanford Cart

• YouTube Video
• Takes 2.6-second for signal to travel from earth to the moon
• Latest iterations is automated with 3D vision capabilities
• Pause after each meter of movement and take 10-15 minutes to reassess its surroundings and reevaluate its decided path.
• In 1979, this cautious version of the cart successfully made its way 20 meters through a chair-strewn room in five hours without human intervention.

What an AI system needs

• Processing of a natural language
• Sensors and actuators
• Representation of knowledge
• Inference from the existing knowledge
• Training on experience (Machine Learning).

Dartmouth workshop

• AI as a science appeared in 1956 at the Dartmouth workshop.
• It was organized by John McCarthy, Marvin Minsky, Claude Shannon and Nathaniel Rochester.
• It was probably the most ambitious grant proposal in the history of computer science.

Dartmouth workshop

From the proposal:

• A 2-month, 10-man study of artificial intelligence, summer 1956, Dartmouth College.
• The conjecture: every aspect of learning or any other feature of intelligence can in principle be so precisely described that a machine can be made to simulate it.
• The agenda: make machines use language, form abstractions and concepts, solve problems now reserved for humans, and improve themselves.
• The bet: a significant advance if a carefully selected group of scientists work on it together for a summer.

1956-1960: Great hopes

• Optimistic time. It seemed a that we were almost there…
• Allen Newell, Herbert A. Simon, and Cliff Shaw: Logic Theorist.
• Automated reasoning.
• It was able to prove most of the Principia Mathematica, in some places even more elegant than Russell and Whitehead.

1956-1960: Big Hopes

• General Problem Solver (Newell and Simon): means-ends analysis. Look at the goal, look at where you are, apply an operator that shrinks the gap. Search, not learning.
• Arthur Samuel’s checkers (1959): the program played itself and got better. No grandmaster wrote the evaluation. AlphaGo is this idea with a neural net.
• McCarthy’s Lisp (1958): the language these programs were written in. Lists as data and as code.
• Microworlds, a few years later: Evans’s Analogy (IQ items), Bobrow’s Student (algebra word problems), Winograd’s Blocks World (stack the red cube). Each worked in a box small enough that “understand” meant a few dozen objects.

1970s: Knowledge Based Systems

• The bottom line: to accumulate a fairly large set of rules and knowledge about the subject area, then draw conclusions.
• First success: MYCIN - Diagnosis of blood infections:
  • about 450 rules
  • The results are like an experienced doctor and significantly better than beginner doctors.

1980-2010: Commercial applications Industry AI

• The first AI department was at DEC (Digital Equipment Corporation). It is argued that by 1986 it saved DEC about $10 million per year.
• The boom ended by the end of the 80s, when many companies could not live up to high expectations.

1990-2010: DATA MINING, MACHINE LEARNING

• In recent decades, the main emphasis has shifted to machine learning and search for patterns in the data.
• Especially - with the development of the Internet.
• Not too many people remember the original AI ideas, but Machine Learning is now everywhere.
• Robotics flourishes (Physical AI) and uses Machine Learning at every step.

Rule-Based System vs Bayes

• Since 1956, the field of artificial intelligence (AI) has undergone significant transformations
• traditional AI was mostly focused on rule-based systems and boolean logic programming, with limited learning capabilities. It lead to them being brittle in changing environments.
• On the other hand, emerging AI is focused on modeling uncertainties, pattern matching, and deep learning.
• All of those are data-driven approaches.
• These approaches are more adaptable and can handle complex and unstructured data. They are also more data-dependent and lack interpretability.

DEFINITION

• How to determine “learning”?

• The definition is very (too?) General.
• What specific examples can be given?

Tasks and concepts of ML

Tasks and concepts of ML: Supervised Learning

• training sample – a set of examples, each of which consists of input features (attributes) and the correct “answers” - the response variable
• Learn a rule that maps input features to the response variable
• Then this rule is applied to new examples (deployment)
• The main thing is to train a model that explains not only examples from the training set, but also new examples (generalizes)
• Otherwise - overfitting

Tasks and concepts of ML: unsupervised learning

There are no correct answers, only data, e.g. clustering:

• We need to divide the data into pre -unknown classes to some extent similar:
  • highlight the family of genes from the sequences of nucleotides
  • cluster users and personalize the application for them
  • cluster the mass spectrometric image to parts with different composition

Tasks and concepts of ML: unsupervised learning

• Dimensionality reduction: a pathologist has 10,000 gene measurements per tumor and 200 tumors. Nobody can look at 10,000 axes. PCA (or a similar map) finds two or three directions that still separate the subtypes. The plot is the product; the 10,000 numbers were the raw material.
• Matrix completion: Netflix is a users-by-movies table that is almost empty. The prize was to fill the holes: what would this person have rated this film? Same job as a sensor that failed for an hour: what would it have read?
• Anomaly detection: a card that usually spends $40 at grocery stores charges $1,200 at 3am in another country. You did not train on a labeled “fraud” class. You trained on what usual looks like, and this is not that.
• Semi-supervised: 10,000 product photos, 200 with labels. Use the unlabeled pile to learn what “looks like the same thing,” then the 200 names to tag the groups. Most of the data never had an answer.

Tasks and concepts of ML: reinforcement learning

• Multi-armed bandits: three headlines for the same landing page. Each visitor is a pull: they click or they do not. You do not know which headline converts. The job is to show ads this week so the total clicks are as high as possible, not to write a report on which headline is best in the limit.
• Exploration vs. exploitation: the headline that is winning so far vs. a fourth headline you have barely tried. Netflix is the same trade: keep recommending the show they already click, or risk a new one that might be the one they actually want.
• Credit assignment: AlphaGo wins 200 moves later. The reward is one bit at the end. Which of those 200 moves deserved it? Same problem as a robot that only hears “the tower stood” after the last block.

Tasks and concepts of ML: active learning

The three paradigms say what you are given. These are decisions you make while you learn. They are not a fourth paradigm.

• Active learning: labels cost money or time. Do not label at random. Query the point where the current model is most unsure, or where an answer would change the decision. A test you have not chosen yet is an expensive experiment.
• Boosting: one weak rule is a coin flip. Many weak rules, each trained on the mistakes of the last, can be a strong classifier. Gradient boosting is this idea with the volume turned up.
• Model selection: a model with more parameters will fit last week’s data. The question is next week’s. Weeks 10 and 11 are this problem: cross-validation, then ridge and lasso as a way to choose the size rather than guess it.
• Ranking: the output is an order, not a class. Search, recommendations, “which patient first.” A good ranking can have a terrible accuracy and still be the right model for the decision.

Tasks and concepts of AI

Tasks and concepts of AI: Reasoning

Reasoning is computing what else follows from what you already believe. Old AI used logic. New AI uses probability. A Bayes net is the clean version. An LLM chain is the messy one. This course is the clean one, so you can check the messy one.

• Bayesian networks: write \(P(\text{child} \mid \text{parents})\). Inference is \(P(\text{query} \mid \text{evidence})\). This is the rest of Module 1.
• Modern loops: ReAct is thought → tool → observation, then again. Tree-of-thought branches and scores paths. Reflexion writes what went wrong and retries. o1-style models hide a search like that inside the model.
• Agents: the same loop is allowed to act (call a function, write a file). The reasoning is in the loop, not in a larger weight matrix.

If you cannot say what was conditioned on, you did not reason. You generated.

Tasks and concepts of AI: Representation

Reasoning needs something to reason over. Generation needs something to generate from. That object is a representation. Old AI wrote it by hand. New AI learns it from data. Same job.

• Knowledge graphs: Paris — capital_of — France. You can query and compose facts. Brittle when the edge was never entered.
• Embeddings: each object is a point. Nearby means similar. “King − man + woman” lands near “queen”. No edge had to be typed.
• Transformers: “Bank” next to “river” is not “bank” next to “loan”. Self-attention builds that context.

A representation is a bet about which distinctions matter. A bad one makes every later model look worse than it is.

Generation: a Bayesian poem

In shadows of data, uncertainty reigns,
Bayesian whispers, where knowledge remains.
With prior beliefs, we start our quest,
Updating with evidence, we strive for the best.

A dance of the models, predictions unfold,
Inferences drawn, from the new and the old.
Through probabilities, we find our way,
In the world of AI, it’s the Bayesian sway.

So gather your data, let prior thoughts flow,
In the realm of the unknown, let your insights grow.
For in this approach, with each little clue,
We weave understanding, both rich and true.

Music

Generation: a DALL-E call

from openai import OpenAI
client = OpenAI(api_key="your-api-key")
response = client.images.generate(
    model="dall-e-3",
    prompt="a hockey player trying to understand the Bayes rule",
    size="1024x1024",
    quality="standard",
    n=1,
)

print(response.data[0].url)

Generation: hockey player

A humorous and illustrative scene of a hockey player sitting on a bench in full gear, holding a hockey stick in one hand and a whiteboard marker in th

Generation: where this actually runs

Where this actually runs. Not a demo.

• Text: email, support replies, code, search answers. ChatGPT, Claude, Gemini, Copilot. A large share of the prose at work this year was proposed by a model.
• Images: product shots, storyboards, the hockey player on the last slide. Midjourney, Flux, Imagen.
• Video ads: Higgsfield takes a product URL and emits UGC-style spots with a consistent spokesperson, native audio, and dozens of variants for Meta and TikTok. Runway, Kling, and Veo do the film-like version of the same job.
• Voice and music: ElevenLabs for speech; Suno for the song two slides back.
• The industrial use in marketing is not one perfect ad. It is a hundred variants, then a test. Generation made the variants cheap. It did not decide which one to run.

Sampling from \(P(\text{output} \mid \text{prompt})\) is the mechanism. Selecting the sample is still a decision.

Chess and AI

Old AI: Deep Blue (1997) vs. Garry Kasparov.

• Search plus a hand-written evaluation. Grandmasters wrote the rules for what a position is worth. The machine did not learn from games.
• About 200 million positions a second. Kasparov lost the match 3.5-2.5. That is brute force, not probability.

Kasparov vs IBM’s DeepBlue in 1997

AlphaGo Zero

• Remove all human knowledge from training process - only uses self play,
• Takes raw board as input and neural network predicts the next move.
• Monte Carlo tree search: from this position, play out many games to the end and count who wins. The move that wins more often is the one you play. That is sampling, not Deep Blue’s 200 million-position brute force.
• The algorithm was able to beat AlphaGo 100-0. The algorithm was then used to play chess and shogi and was able to beat the best human players in those games as well.

Alpha GO vs Lee Sedol: Move 37 by AlphaGo in Game Two

Probability in machine learning

A model that only emits a point answer is unfinished. How confident is this? How well does it describe the data we have? If we ran the experiment again, how much would the number move? Classification, generation, and AlphaGo all produce those questions. A rule that fires or does not cannot.

• Supervised: \(P(y \mid x)\), not only \(\hat{y}\).
• Unsupervised: a distribution over structure we did not label.
• Reinforcement: random rewards; exploration is a bet about what we do not know yet.
• Generation: sampling from \(P(\text{output} \mid \text{prompt})\).
• Old AI: if rain, take umbrella. New AI: \(P(\text{umbrella} \mid \text{rain})\), which you can invert.

The rest of tonight, and the next six weeks, is the machinery behind those statements.

Review of Basic Probability Concepts

Probability lets us talk efficiently about things that we are uncertain about.

• What will Amazon’s sales be next quarter?
• What will the return be on my stocks next year?
• How often will users click on a particular Google ad?

All these involve estimating or predicting unknowns!!

Random Variables

Random Variables are numbers that we are not sure about. There’s a list of potential outcomes. We assign probabilities to each outcome.

Example: Two fair coins. Four equally likely outcomes:

\[\{TT,\; TH,\; HT,\; HH\}\]

Let \(X\) be the number of heads. \(X\) is not the coins. \(X\) is a function from those four outcomes to a number.

Random Variables

\[ \begin{array}{c|c} \text{outcome} & X \\ \hline TT & 0 \\ TH & 1 \\ HT & 1 \\ HH & 2 \end{array} \]

So \(X\) can be \(0\), \(1\), or \(2\). We do not yet know which. The next slide is the list of probabilities for those three numbers. That list is the distribution of \(X\).

Probability Distribution

We describe the behavior of random variables with a Probability Distribution

Example: Suppose we are about to toss two coins. Let \(X\) denote the number of heads.

\[X = \left\{ \begin{array}{ll} 0 \text{ with prob. } 1/4\\ 1 \text{ with prob. } 1/2\\ 2 \text{ with prob. } 1/4 \end{array} \right.\]

\(X\) is called a Discrete Random Variable

Question: What is \(P(X=0)\)? How about \(P(X \geq 1)\)?

Pete Rose Hitting Streak

Pete Rose of the Cincinnati Reds set a National League record of hitting safely in \(44\) consecutive games …

• Rose was a \(300\) hitter. Assume he comes to bat \(4\) times each game.
• Each at bat is assumed to be independent, i.e., the current at bat doesn’t affect the outcome of the next.

What probability might reasonably be associated with that hitting streak?

Joe DiMaggio’s record is \(56\)! His batting average was \(.325\)

Pete Rose Hitting Streak (Solution)

Let \(A_i\) be the event that Rose hits safely in game \(i\). Independence across games gives

\[P(A_1 \cap \cdots \cap A_{44}) = P(A_1)\cdots P(A_{44}) = \bigl(P(A_1)\bigr)^{44}\]

One game, four independent at-bats, batting average \(0.300\):

\[\begin{align*} P(A_1) &= 1 - P(\text{four outs}) \\ &= 1 - (0.7)^4 \\ &= 0.7599 \end{align*}\]

So \[P(\text{44-game hitting streak}) = (0.7599)^{44} \approx 5.67 \times 10^{-6}\]

Pete Rose Hitting Streak (Inference)

There are three basic inferences:

• The odds for a particular player as good as Pete Rose starting a hitting streak today are about 176,500 to 1
• That does not mean the run of \(44\) will never be beaten by some player at some time: the Law of Very Large Numbers
• Joe DiMaggio’s record is \(56\). Batting average \(0.325\), so \(P(A_i)=1-(0.675)^4=0.7924\), and \((0.7924)^{56} \approx 2.19 \times 10^{-6}\), or about 456,000 to 1. Hard to beat.

Conditional, Joint and Marginal Distributions

Use probability to describe outcomes involving more than one variable at a time. Need to be able to measure what we think will happen to one variable relative to another

In general the notation is ...

• \(P(X=x, Y=y )\) is the joint probability that \(X =x\) and \(Y=y\)
• \(P(X=x \mid Y=y )\) is the conditional probability that \(X\) equals \(x\) given \(Y=y\)
• \(P(X=x)\) is the marginal probability of \(X=x\)

Conditional, Joint and Marginal Distributions

Relationship between the joint and conditional ...

\[\begin{align*} P(x,y) & = P(x) P(y \mid x) \\ & = P(y) P(x \mid y) \end{align*}\]

Relationship between the joint and marginal ...

\[\begin{align*} P(x) & = \sum_y P(x,y) \\ P(y) & = \sum_x P(x,y) \end{align*}\]

Example: Happiness Index

“happiness index” as a function of salary.

Is \(P(Y=2 \mid X=3) > P(Y=2)\)?

Bayes Rule: from joint and prior

The computation of \(P(x \mid y)\) from \(P(x)\) and \(P(y \mid x)\) is called Bayes theorem ... \[ P(x \mid y) = \frac{P(y,x)}{P(y)} = \frac{P(y,x)}{\sum_x P(y,x)} = \frac{P(y \mid x)P(x)}{\sum_x P(y \mid x)P(x)} \]

This shows now the conditional distribution is related to the joint and marginal distributions.

You’ll be given all the quantities on the r.h.s.

Bayes Rule: \(P(x \mid y)\) is not \(P(y \mid x)\)

Key fact: \(P(x \mid y)\) is generally different from \(P(y \mid x)\)!

Example: Most people would agree

\[\begin{align*} Pr & \left ( Practice \; hard \mid Play \; in \; NBA \right ) \approx 1\\ Pr & \left ( Play \; in \; NBA \mid Practice \; hard \right ) \approx 0 \end{align*}\]

The main reason for the difference is that \(P( Play \; in \; NBA ) \approx 0\).

Independence

Two random variable \(X\) and \(Y\) are independent if \[ P(Y = y \mid X = x) = P (Y = y) \] for all possible \(x\) and \(y\) values. Knowing \(X=x\) tells you nothing about \(Y\)!

Example: Tossing a coin twice. What’s the probability of getting \(H\) in the second toss given we saw a \(T\) in the first one?

Sally Clark Case: Independence or Bayes?

Sally Clark was accused and convicted of killing her two children

They could have both died of SIDS.

• The chance of a family which are non-smokers and over 25 having a SIDS death is around 1 in 8,500.

• The chance of a family which has already had a SIDS death having a second is around 1 in 100.

• The chance of a mother killing her two children is around 1 in 1,000,000.

Random Variables: Expectation \(E(X)\)

The expected value of a random variable is simply a weighted average of the possible values X can assume.

The weights are the probabilities of occurrence of those values.

\[E(X) = \sum_x xP(X=x)\]

With \(n\) equally likely outcomes with values \(x_1, \ldots, x_n\), \(P(X = x_i) = 1/n\)

\[E(X) = \frac{x_1+x_2+\ldots+x_n}{n}\]

Roulette Expectation

• European Odds: 36 numbers (red/black) + zero
• You bet $1 on 11 Black (pays 35 to 1)
• \(X\) is the return on this bet

\[E(X) = \frac{1}{37}\times 36 + \frac{36}{37}\times 0 = 0.97\]

• If you bet $1 on Black (pays 1 to 1)

\[E(X) = \frac{18}{37}\times 2 + \frac{19}{37}\times 0 = 0.97\]

Casino is guaranteed to make money in the long run!

Standard Deviation \(sd(X)\) and Variance \(Var(X)\)

The variance is calculated as

\[Var(X) = E\left((X - E(X))^2\right)\]

A simpler calculation is \(Var(X) = E(X^2) - E(X)^2\).

The standard deviation is the square-root of variance.

\[sd(X) = \sqrt{Var(X)}\]

Roulette Variance

• European Odds: 36 numbers (red/black) + zero
• You bet $1 on 11 Black (pays 35 to 1)
• \(X\) is the return on this bet

\[Var(X) = \frac{1}{37}\times (36 - 0.97)^2 + \frac{36}{37}\times (0 - 0.97)^2 = 34\]

• If you bet $1 on Black (pays 1 to 1)

\[Var(X) = \frac{18}{37}\times (2 - 0.97)^2+ \frac{19}{37}\times (0- 0.97)^2 = 1\]

If your goal is to spend as much time as possible in the casino (free drinks): place small bets on black/red

Example: \(E(X)\) and \(Var(X)\)

Tortoise and Hare are selling cars. Probability distributions, means and variances for \(X\), the number of cars sold

Expectation and Variance Calculations

Let’s do Tortoise expectations and variances

• The Tortoise \[\begin{align*} E(T) &= (1/2)(1) + (1/2)(2) = 1.5 \\ Var(T) &= E(T^2) - E(T)^2 \\ &= (1/2)(1)^2 + (1/2)(2)^2 - (1.5)^2 = 0.25 \end{align*}\]

• Now the Hare’s \[\begin{align*} E(H) &= (1/2)(0) + (1/2)(3) = 1.5 \\ Var(H) &= (1/2)(0)^2 + (1/2)(3)^2- (1.5)^2 = 2.25 \end{align*}\]

Expectation and Variance Interpretation

What do these tell us above the long run behavior?

• Tortoise and Hare have the same expected number of cars sold.
• Tortoise is more predictable than Hare. He has a smaller variance The standard deviations \(\sqrt{Var(X)}\) are \(0.5\) and \(1.5\), respectively
• Given two equal means, you always want to pick the lower variance.

Odds: Oddschecker

We can express probabilities in terms of Odds via \[ O(A) = \frac{ 1- P(A) }{ P(A) } \; \; \text{or} \; \; P(A) = \frac{ 1 }{ 1 + O(A) } \]

• For example if \(O(A) = 1\) then for ever $1 bet you will payout $1. An event with probability \(\frac{1}{2}\).
• If \(O(A) = 2\) or \(2:1\), then for a $1 bet you’ll payback $3.

In terms of probability \(P = \frac{1}{3}\).

Bookies vs Betters: The Battle of Probabilistic Models

Source: The Secret Betting Strategy That Beats Online Bookmakers

Bookies vs Betters: The Battle of Probabilistic Models

• Bookies set odds that reflect their best guess on probabilities of a win, draw, or loss. Plus their own margin
• Bookies have risk aversion bias. When many people bet for an underdog (more popular team)
• Bookies hedge their bets by offering more favorable odds to the opposed team
• Simple algorithm: calculate average odds across many bookies and find outliers with large deviation from the mean

Covariance

Suppose that we have two random variables \(X\) and \(Y\)

We need to measure whether they move together or in opposite directions The Covariance is defined by

\[Cov(X,Y) = E\left((X- E(X))(Y- E(Y))\right)\]

In terms of probability distributions, we need to calculate

\[Cov(X,Y) = \sum_{x,y} (x - E(X))(y - E(Y))P(x,y)\]

Let’s look at Covariance on Markets

VIX vs S&P vs AAPL

Correlation

The Correlation is defined by

\[Corr(X,Y) = \frac{Cov(X,Y)}{sd(X) sd(Y)}\]

• What are the units of \(Corr(X,Y)\)? They don’t depend on the units of \(X\) or \(Y\)!
• \(-1 \leq Corr(X,Y) \leq 1\)

If \(Cov(\text{Apple Sales, Economy}) = 5\), \(sd(\text{Apple Sales}) = 2\) and \(sd(\text{Economy}) = 3.5\), then there’s a \(71.4\)% correlation

\[Corr(\text{Apple Sales, Economy}) = \frac{5}{2 \times 3.5} = \frac{5}{7} = 0.714\]

Linear Combinations of Random Variables

Two key properties:

Let \(a, b\) be given constants

• Expectations and Variances \[\begin{align*} E(aX + bY) &= a E(X) + b E(Y) \\ Var(aX + bY) &= a^2 Var(X) + b^2 Var(Y) + 2 ab Cov(X,Y) \end{align*}\]

where \(Cov(X,Y)\) is the covariance between random variables.

Medical Diagnostics - Visualization

Bayes Rule - Intuitive Explanation

Let’s think about this intuitively. Imagine 1000 people like Alice.

• 10 have cancer, 990 do not.
• 80% of those with cancer test positive: 8.
• 10% of those without cancer test positive: 99.
• In total, 107 positive mammograms. That 107 is the law of total probability: positives from cancer plus positives from no cancer.

Now pick one of those 107 at random. What is the chance she has cancer? \[P(\text{cancer} \mid \text{positive}) = \frac{8}{107} \approx 7.5\%\]

Calculation of Posterior Probability

\[P(\text{cancer} \mid \text{positive mammogram})=\frac{8}{107}\approx 7.5\%\]

Posterior Probability

Most women who test positive on a mammogram are healthy, because the vast majority of women who receive mammograms in the first place are healthy.

Apple Watch

The Apple Watch Series 4 can perform a single-lead ECG and detect atrial fibrillation. The software can correctly identify 98% of cases of atrial fibrillation (sensitivity) and 99% of cases of non-atrial fibrillation (specificity).

Take prevalence \(P(\text{AF})=0.02\) in the population we care about. What is \(P(\text{AF} \mid \text{watch says AF})\)?

Bayes’ Theorem: \[ P(A|B)=\frac{P(B|A)P(A)}{P(B)} \]

Apple Watch

Counts from 100,000 people at 2% prevalence, 98% sensitivity, 99% specificity:

\[ 0.6667 = \frac{0.98\cdot 0.02}{ 0.0294} \]

The conditional probability of having atrial fibrillation when the Apple Watch Series 4 detects atrial fibrillation is about 67%.

“Personalization" \(=\)”Conditional Probability"

• Conditional probability is how AI systems express judgments in a way that reflects their partial knowledge.
• Personalization runs on conditional probabilities, all of which must be estimated from massive data sets in which you are the conditioning event.

Many Business Applications!! Suggestions vs Search….

Probability as Evidence

evidence: known facts about criminal (e.g. blood type, DNA, ...)

suspect: matches a trait with evidence at scene of crime

Let \(G\) denote the event that the suspect is the criminal.

Bayes computes the conditional probability of guilt

\[ P ( G | \text{evidence} ) \] Evidence \(E\): suspect and criminal possess a common trait

Probability as Evidence

Bayes Theorem yields \[ P ( G | \text{evidence} ) = \frac{ P ( \text{evidence} | G ) P ( G ) }{ P ( \text{evidence} )} \]

In terms of relative odds \[ \frac{ P ( I | \text{evidence} ) }{ P ( G | \text{evidence} ) } = \frac{ P ( \text{evidence} | I ) }{ P ( \text{evidence} | G ) } \frac{ P ( I ) }{ P ( G ) } \]

Prosecutor’s Fallacy

The most common fallacy is confusing \[ P ( \text{evidence} | G ) \; \; \text{with} \; \; P ( G | \text{evidence} ) \]

Bayes rule yields \[ P ( G | \text{evidence} ) = \frac{ P ( \text{evidence} | G ) p( G )}{ P ( \text{evidence} )} \] Your assessment of \(P( G )\) will matter.

Island Problem

Suppose there’s a criminal on an island of \(N+1\) people.

• Let \(I\) denote innocence and \(G\) guilt of a particular suspect.
• Evidence \(E\): the suspect matches a trait found at the scene. The trait has frequency \(p\) in the population.
• Then \[ p(E \mid I)=p,\qquad p(E \mid G)=1 \]

Bayes factor

Bayes factors are likelihood ratios

• The Bayes factor is given by \[ \frac{p(E|I)}{p(E|G)}=p \]

• Uniform prior over the \(N+1\) people: \[ p(G)=\frac{1}{N+1},\qquad p(I)=\frac{N}{N+1},\qquad odds(I)=N \]

• Posterior odds of innocence: \(odds(I \mid E) = Np\). Priors will matter.

Kramnik vs Nakamura

November 2023. Vladimir Kramnik, world champion 2000–2007, writes that Hikaru Nakamura’s 45.5/46 in 3+0 blitz on chess.com looks like cheating. The window is cherry-picked from more than 3,500 games.

Kramnik’s table: 45.5/46 against opponents around 2950. Implied performance 3627. Nakamura’s rating is 3300.

Kramnik’s argument: that streak is rare, so maybe he cheated. That is \(P(E \mid I)\) small. We want \(P(I \mid E)\). Same machine as the island.

Kramnik vs Nakamura: Anand’s prior

Kramnik vs Nakamura: the streak

Nakamura 3300, opponents 2934. A 400-point Elo gap is 10-to-1. Difference 366: \[ q = \frac{10^{366/400}}{1+10^{366/400}} = 0.8916 \] Forty-six places for the one non-win: \[ p = P(E \mid I) = 46 \times 0.8916^{45} \times 0.1084 = 0.0286 \] About 3%. Not one in a million.

Kramnik vs Nakamura: posterior

\(N=10{,}000\) and \(p=0.0286\), so \(odds(I \mid E)=Np=286\) and \[ P(I \mid E) = \frac{286}{287} \approx 99.65\% \]

The streak is a 3% event under innocence. Cheating, under Anand, is a 0.01% event. Kramnik quoted the 3%.

Even if cheating is 20 times more common than Anand says (\(N=500\)), \(P(I \mid E)\) is still 93.5%.

OJ Simpson

The O.J. Simpson trial was possibly the trial of the century

The murder of his wife Nicole Brown Simpson, and a friend, Ron Goldman, in June 1994 and the trial dominated the TV networks

• DNA evidence and probability: \(p( E| G)\)

• Bayes Theorem: \(p( G | E )\)

• Prosecutor’s Fallacy: \(p( G|E ) \neq p(E|G)\)

Odds ratio \[ \frac{ p( I|E) }{ p ( G | E ) } = \frac{ p( E|I )}{ p( E|G) } \frac{ p(I) }{p(G ) } \] Prior odds conditioned on background information.

OJ Simpson: Bayes Theorem

• Only \(1/10\)th of one percent of husbands who batter their wife murder them in a given year, so \(p(G\mid B)=1/1000\).

In 1994, 5000 women were murdered, 1500 by their husband

Given a population of 100 million women at the time \[ p( M | I ) = \frac{ 3500 }{ 10^8 } \approx \frac{1}{30,000} . \] We’ll also need \(p( M | I , B ) = p( M | I )\)

OJ Simpson: Prosecutor’s Fallacy

• Let \(G =\) Guilt and \(E=\) Evidence
• Prosecutor’s Fallacy: \(P(G|E) \neq P(E|G)\).
• DNA evidence gives \(P( E | I )\) – the \(p\)-value.

What’s the “match probability” for a rare event?

Bayes theorem in Odds \[ \frac{p(G|M,B)}{p(I|M,B)} = \frac{p(M|G,B)}{p(M|I,B)} \frac{p(G|B)}{p(I|B)} \]

Base Rate Fallacies

“Witness” 80 % certain saw a “checker” \(C\) taxi in the accident.

• What’s your \(P ( C | E )\) ?

• Need \(P ( C )\). Say \(P( C ) = 0.2\) and \(P( E | C) = 0.8\).

• Then your posterior is

\[ P ( C | E ) = \frac{0.8 \cdot 0.2}{ 0.8 \cdot 0.2 + 0.2 \cdot 0.8 } = 0.5 \]

Therefore \(O ( C ) = 1\) a 50/50 bet.

Envelope Paradox

The following problem is known as the “exchange paradox”.

• A swami puts \(m\) dollars in one envelope and \(2 m\) in another. He hands on envelope to you and one to your opponent.
• The amounts are placed randomly and so there is a probability of \(\frac{1}{2}\) that you get either envelope.
• You open your envelope and find \(x\) dollars. Let \(y\) be the amount in your opponent’s envelope.

Envelope Paradox

You know that \(y = \frac{1}{2} x\) or \(y = 2 x\). You are thinking about whether you should switch your opened envelope for the unopened envelope of your friend. It is tempting to do an expected value calculation as follows \[ E( y) = \frac{1}{2} \cdot \frac{1}{2} x + \frac{1}{2} \cdot 2 x = \frac{5}{4} x > x \] Therefore, it looks as if you should switch no matter what value of \(x\) you see. A consequence of this, following the logic of backwards induction, that even if you didn’t open your envelope that you would want to switch!

Three Prisoners Problem

Three prisoners \(A , B , C\).

Each believe are equally likely to be set free.

Prisoner \(A\) goes to the warden \(W\) and asks if s/he is getting axed.

• The Warden can’t tell \(A\) anything about him.

• He provides the new information: \(WB\) = “\(B\) is to be executed”

Game Show Problem

Named after the host of the long-running TV show, Let’s make a Deal.

• A contestant is given the choice of 3 doors.

There is a prize (a car, say) behind one of the doors and something worthless behind the other two doors: two goats.

• The optimal strategy is counter-intuitive

Puzzle

The game is as follows:

• You pick a door.

• Monty then opens one of the other two doors, revealing a goat.

• You have the choice of switching doors.

Is it advantageous to switch?

Assume you pick door \(A\) at random. Then \(P(A) = ( 1 /3 )\).

You need to figure out \(P( A | MB )\) after Monte reveals \(B\) is a goat.

Bayes Rule Meme

Sheldon Bayes

The Reverend and the Submarine

• A missing submarine. A car in snow that has to know where it is. Same update.
• Start with a prior over the map. Look. Move the probability to where the data now say.
• John Craven did this for USS Scorpion. Larry Stone at Metron turned that hunt into a method you can run again: prior map, look, revise.
• A self-driving stack does the same update every few milliseconds. That loop is SLAM: simultaneous localization and mapping.

\[P(H \mid D)=\frac{P(H)\,P(D\mid H)}{P(D)}\]

The Story of Scorpion

• May 1968. USS Scorpion and 99 men are overdue, somewhere in thousands of miles of Atlantic.
• The Pentagon calls John Craven. He does not sweep the ocean. He asks experts to bet on what happened: torpedo, battery, collision.
• Those bets become a prior map. Search the hottest square. Find nothing: take probability off that square and put it elsewhere.
• They found the wreck. The method was already old. The Allies had used it on U-boats.

Bayesian Search

• Same playbook, later wrecks.
• Air France 447: two years of dragging. Then a Bayesian map. Found in weeks.
• MH370: the search itself is written up as this update (Davey et al.).
• Combine every source into a prior. Look. Revise. Repeat.
• Prior + facts = where to look next.

Sensitivity and Specificity

Two errors: An infected person may test negative, a well person tests positive.

Sensitivity (or power) \(=\) true positive rate (or recall) % sick people who are correctly identified \(P(T \mid D)\).

In a perfect world, we’d like \(P(\bar{T} \mid D) \approx 0\)

Specificity \(=\) true negative rate % of negatives correctly identified as such \(P(\bar{T} \mid \bar{D})\).

• False negative rate \(=\) 1 - sensitivity \(=\) \(\beta\), the type II error
• False positive rate \(=\) 1 - specificity \(=\) \(\alpha\), the type I error

We want the probability: \(P(D \mid T)\)

Test Marketing a New Product

Your company will test-market a new product. After a positive, neutral, or negative reaction, what are the probabilities of high and low sales?

• New products have high sales 8% of the time, low sales 92%.
• If sales are high: reaction is positive 70%, neutral 25%, negative 5%.
• If sales are low: positive 15%, neutral 35%, negative 50%.

Netflix Bayes and AI

Test Marketing - Solution Steps

Step 1: Set-up your notation. Let \[H = \text{high sales} \quad L = \text{low sales}\] \[\text{Pos} = \text{positive} \quad \text{Neu = Neutral} \quad \text{Neg} = \text{Negative}\]

Step 2: List the known conditional probabilities For the marketing test we have \[\begin{align*} P(\text{Pos} \mid H) &= 0.70, P(\text{Neu} \mid H)=0.25, P(\text{Neg} \mid H) = 0.05 \\ P(\text{Pos} \mid L) &= 0.15, P(\text{Neu} \mid L)=0.35, P(\text{Neg} \mid L) = 0.50 \end{align*}\]

Finally, the base rates are \(P(H) = 0.08\) and \(P(L) = 0.92\)

Test Marketing - Calculation

Step 3: Describe the posterior probabilities that are required: \[P(H \mid \text{Pos})\]

The probability of high sales given a positive marketing test. Compute the probability of a positive test \[\begin{align*} P(\text{Pos}) &= P(\text{Pos} \mid H)P(H) +P(\text{Pos} \mid L)P(L) \\ &= 0.70 \times 0.08 + 0.15 \times 0.92 = 0.194 \end{align*}\]

Test Marketing - Final Answer

Now use Bayes Rule \[\begin{align*} P(H \mid \text{Pos}) &= \frac{P(\text{Pos} \mid H)P(H)}{P(\text{Pos})} \\ &= \frac{0.70 \times 0.08}{0.194} = 0.288 \end{align*}\]

Hence 28.8% you’ll have high sales in the market.

We should interpret this relative to our initial probability of only \(8\)%.

Two Headed Coin

Large jar containing \(1024\) fair coins and one two-headed coin.

• You pick one at random and flip it \(10\) times and get all heads.
• What’s the probability that the coin is the two-headed coin?

\(\frac{1}{1025}\) probability of initially picking the two headed coin. \(\frac{1}{1024}\) chance of getting \(10\) heads in a row from a fair coin Therefore, it’s a \(50/50\) bet.

Two Headed Coin - Calculation

Let \(E\) be the event that you get \(10\) Heads in a row \[P(\text{two headed} \mid E) = \frac{P(E \mid \text{two headed})P(\text{two headed})}{P(E \mid \text{fair})P(\text{fair}) + P(E \mid \text{two headed})P(\text{two headed})}\]

Therefore, the posterior probability \[P(\text{two headed} \mid E) = \frac{1 \times \frac{1}{1025}}{\frac{1}{1024} \times \frac{1024}{1025} + 1 \times \frac{1}{1025}} = 0.50\]

Bayes Classifier

Input \(X\) and label \(Y\):

\(X:\) the image \(Y:\) the label, “cat” or not

Silicon Valley: Season 4: Not Hotdog

What’s our best decision?

Pick \(\hat{y}\) as the most likely category given that \(X=x\), namely \[\hat{y} = \arg \max_y \; p(Y= y \mid X = x)\]

I need the probability table \(P(X=x, Y=y)\) and marginal \(P(X=x)\).

Naive Bayes

Use of the Bayes rule allows us to build our first predictive model, called Naive Bayes classifier.

• Collection of classification algorithms based on Bayes Theorem
• Every feature being classified is independent of any other feature
• Features aren’t always independent → “naive” assumption

Naive Bayes: Example

A fruit may be considered an apple if it is:

• Red, round, and about 3” in diameter

Each feature contributes independently to the probability:

graph TB
    C((Color)) --> F((Fruit))
    S((Size)) --> F
    Sh((Shape)) --> F

graph TB
    C((Color)) --> F((Fruit))
    S((Size)) --> F
    Sh((Shape)) --> F

Naive Bayes: Advantages

• Relatively simple to understand and build
• Easily trained, even with small datasets
• Fast execution
• Not sensitive to irrelevant features

Naive Bayes: Fruit Classification Example

Naive Bayes: Classification

Given evidence \(E\) (Long, Sweet, Yellow), calculate probability for each class:

Banana: \[P(B \mid E) = \frac{0.8 \times 0.7 \times 0.9 \times 0.5}{P(E)} = \frac{0.252}{P(E)}\]

Orange: \[P(O \mid E) = 0\] (since P(Long|Orange) = 0)

Other Fruit: \[P(F \mid E) = \frac{0.5 \times 0.75 \times 0.25 \times 0.2}{P(E)} = \frac{0.01875}{P(E)}\]

Result: Long, Sweet, Yellow fruit is classified as a Banana

Naive Bayes: Spam Filtering

The original spam filtering algorithm was based on Naive Bayes.

Key Assumptions:

• Words are conditionally independent given the class label
• Ignores word order and grammatical relationships
• Treats each word independently

flowchart TB
  w1(w1)-->S(S)
  w2(w2)-->S
  w3(w3)-->S
  w4(w4)-->S
  w5(w5)-->S
  w6(w6)-->S
  w7(w7)-->S

flowchart TB
  w1(w1)-->S(S)
  w2(w2)-->S
  w3(w3)-->S
  w4(w4)-->S
  w5(w5)-->S
  w6(w6)-->S
  w7(w7)-->S

Naive Bayes: Spam Classification

For a new email with words \(\{w_{e1}, w_{e2}, \ldots, w_{ek}\}\):

\[P(\text{spam} \mid \text{email}) = \frac{P(\text{email} \mid \text{spam}) \times P(\text{spam})}{P(\text{email})}\]

Where: \[P(\text{email} \mid \text{spam}) = P(w_{e1} \mid \text{spam}) \times P(w_{e2} \mid \text{spam}) \times \ldots \times P(w_{ek} \mid \text{spam})\]

Naive Bayes: Limitations

The “naive” assumption:

• Words are not truly independent
• Ignores grammar and word order
• Doesn’t understand context (e.g., “bank” as financial vs. river)

Despite limitations:

• Often performs well in practice
• Excellent for text classification tasks
• Computational efficiency

Naive Bayes: Modern Context

While Naive Bayes has limitations, it paved the way for more sophisticated models:

• Transformers consider sequential nature of language
• Contextual embeddings capture word relationships
• Deep learning models understand grammar and syntax

But Naive Bayes remains valuable for:

• Quick prototyping
• Baseline models
• Resource-constrained applications

Alarm: the CPT

Alarm: the DAG

graph TB
    b((b)) --> a((a))
    e((e)) --> a
    e --> r((r))

graph TB
    b((b)) --> a((a))
    e((e)) --> a
    e --> r((r))

Alarm: the numbers

\[ p(a=1 \mid b,e) + p(a=0 \mid b,e) = 1. \] Also know \(p(r=1 \mid e=1) = 0.5\) and \(p(r=1 \mid e=0) = 0\), \(p(b) = 2\cdot10^{-4}\) and \(p(e) = 10^{-2}\) (historical data)

Graph allowed us to have a more compact representation of the joint probability distribution. The original naive representations requires specifying \(2^4\) parameters.

Alarm: a higher crime rate

• We get such a low probability of burglary because its prior is very low compared to prior probability of an earthquake.
• What will happen to posterior if we live in an area with higher crime rates, say \(p(b) = 10^{-3}\). \[ p(b \mid a) = \dfrac{p(b)}{p(b) + 10^{-3}(1-p(b))} \]

Alarm: prior vs posterior

prior <- seq(0, .1, length.out = 200)
post <- prior / (prior + 0.001 * (1 - prior))
plot(prior, post, type = "l", lwd = 3, col = "red")

Figure 3: Relationship between the prior and posterior

Bayes and AI

Translate a sentence. Hear a wake word. Stay in the lane. Pick the next show.

None of those is certain. Each is the question you have been answering all night: given what I just saw, what is likely?

• An AI algorithm deals in probabilities, not certainties.
• It does not arrive with the instructions. It updates them: prior, data, posterior.

That is Bayes. Next: will this subscriber like Saving Private Ryan?

How does Netflix Give Recommendations?

Will a subscriber like Saving Private Ryan, given that he or she liked the HBO series Band of Brothers?

Both are epic dramas about the Normandy invasion and its aftermath.

100 people in your database, and every one of them has seen both films.

Their viewing histories come in the form of a big “ratings matrix”.

\[P(\text{likes Saving Private Ryan} \mid \text{likes Band of Brothers})=\frac{56}{56+14}=80\%\]

How does Netflix Give Recommendations? - Fundamental Equation

The fundamental equation is: \[\text{Predicted Rating} =\text{Overall Average} + \text{Film Offset} + \text{User Offset} + \text{User-Film Interaction}\]

These three terms provide a baseline for a given user/film pair:

• The overall average rating across all films is 3.7.
• Every film has its own offset. Popular movies have positive offsets.
• Every user has an offset. Some users are more or less critical than average.

Netflix - Latent Features

The leftover term, user-film interaction, is not a one-off. Ratings of similar shows move together because they share a hidden feature of the person.

• You never observe the feature. You infer it from what they rated.
• There are not one but dozens or hundreds: British murder mystery, gritty crime drama, cooking show, hipster comedy.
• A film has a vector of how much it has each feature. A person has a vector of how much they like each feature.
• The interaction is how well those two vectors line up. That is why two people with the same average rating still get different recommendations.

The Hidden Features Tell the Story

• Those vectors put you in a corner of a high-dimensional space. That is a demographic of one, not an age-and-zip-code bin.
• Netflix spent $130 million on 10 episodes of The Crown because the model said who would watch.
• Network television spent $400 million on 113 pilots. Thirteen shows made a second season.

New England Patriots and Coin Tossing

Patriots won 19 out of 25 coin tosses in 2014-15 season. What is the probability of that happening?

• Each toss is a Bernoulli trial: \(X=1\) if they win, \(P(X=1)=\tfrac12\). Then \(Y\sim\mathrm{Binomial}(25, \tfrac12)\).
• There are \(25\) choose \(19\), or \(177{,}100\), sequences with 19 wins. Each has probability \(0.5^{25}\).

\[\Pr(Y=19) = {25 \choose 19} 0.5^{25} = 177{,}100 \times 0.5^{25} = 0.005\]

Rare under a fair toss. Not a proof they cheated.

Binomial Distribution Formula

EPL Odds

EPL Odds

EPL 2017 Data

We have a historical set of data on scores

Tomorrow Manchester United (MU) is playing Hall U. I want to place a bet.

How can I predict the outcome of this game?

Poisson Distribution

The Poisson distribution counts the occurrence of events Given the rate \(\lambda\) we calculate probabilities as follows

\[P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} \quad \text{where } x=0,1,2,3, \ldots\]

The Poisson Mean and Variance are:

\(\lambda\) is the rate of occurrence of an event.

EPL MU

I can build a model assuming goals follow Poisson distribution.

I calculate \(\lambda\) by taking an average.

Our Poisson model fits the empirical data!!

English Premier League: EPL

Calculate Odds for the possible scores in a match?

\[0-0, \; 1-0, \; 0-1, \; 1-1, \; 2-0, \ldots\]

Let

\(X=\) Goals scored by Hall U

\(Y=\) Goals scored by MU

What’s the odds of a MU winning? \(P(X < Y)\) Odds of a draw? \(P(X = Y)\)

x = rpois(100,0.6)
y = rpois(100,1.4)
sum(x<y)/100   # Team 2 wins
sum(x==y)/100   # Draw

EPL: Attack and Defence Strength

Each team gets an “attack” strength and “defence” weakness rating Adjust home and away average goal estimates

EPL Table

EPL: Hull vs ManU

ManU (away). League away average \(1.20\) from epl.csv. Prediction: \(1.20 \times 1.46 \times 1.37 = 2.41\)

Average \(\times\) Attack strength \(\times\) Defense weakness

Hull (home). League home average \(1.60\). Prediction: \(1.60 \times 0.85 \times 0.52 = 0.71\).

The two league averages are different. Using one number for both would erase home advantage, which is the effect this model is built to capture.

Simulation (\(n=100\))

EPL Predictions

A model is only as good as its predictions

• In our simulation Man U wins 72 games out of 100, we should bet when the quoted odds are worse than that.
• Most likely Man U score in the simulation is 2.
• The actual outcome was 0-1 (they played on 27 August 2016).
• In our simulation 0-1 occurred 11 times out of 100.

Continuous Random Variables

Suppose we are trying to predict tomorrow’s return on the S&P500…

There’s a number of questions that come to mind

• What is the random variable of interest?
• How can we describe our uncertainty about tomorrow’s outcome?
• Instead of listing all possible values we’ll work with intervals instead. The probability of an interval is defined by the area under the probability density function.

They are continuous (as opposed to discrete) random variables

Normal Distribution

There are infinitely many normals: every mean, every variance. They are all the same bell, just shifted and stretched. We keep one copy.

• \(Z\sim N(0,1)\): mean \(0\), variance \(1\). That is the table.
• \(P(|Z|<1)\approx 0.68\). \(P(|Z|<1.96)\approx 0.95\). Those numbers are the whole family, once you measure in units of \(\sigma\).
• R: pnorm, qnorm, rnorm(1000, 0, 1).

The Normal Distribution - Standardization

Our probability model is written \(X \sim N(\mu,\sigma^2)\) \(\mu\) is the mean, \(\sigma^2\) is the variance

• Standardization if \(X \sim N(\mu,\sigma^2)\) then \[Z =\frac{X-\mu}{\sigma} \sim N(0,1)\]
• \(\mu:\) the center of the distribution \(\sigma:\) how spread out the data are

\(95\)% probability \(X\) is inside \(\mu \pm 1.96 \sigma\).

Normal Distribution - Mean Parameter

By changing the mean parameter \(\mu\), we change the center of the bell curve

Normal Distribution - Mean

Normal Distribution - Variance Parameter

By changing the variance parameter \(\sigma^2\), we change the “fatness” of the bell curve

Normal Distribution - Variance

Normal Distribution - Real Data

Chicago Wind Speed (2007-2014) data on a log scale seem to be well described by the Normal distribution

Normal Distribution - Wind Data

pnorm and qnorm

We can find probabilities and quantiles in R. Here are the important values

>pnorm(2.58)
[1] 0.9950
>pnorm(1.96)
[1] 0.9750
>pnorm(1.64)
[1] 0.9499

qnorm is the inverse of pnorm. Simulation rnorm. N=1000, x=rnorm(N,0,1), p=sum(x<1.96)/N

Normal Distribution with General Mean and Variance

Here are two useful facts: If \(X \sim \text{N}(\mu, \sigma^2)\), then

\[\begin{align*} P(\mu - 2.58 \sigma < X < \mu + 2.58 \sigma) =& 0.99 \\ P(\mu - 1.96 \sigma < X < \mu + 1.96 \sigma) =& 0.95 \end{align*}\]

• The chance that \(X\) will be within \(2.58 \sigma\) of its mean is \(99\)%, and the chance that it will be within \(2\sigma\) of its mean is about \(95\)%.

Pictorially

Normal Distribution Tails

Examples of upper and lower tail areas. The lower tail area of \(0.1\) is at \(z = -1.28\). The upper tail area of \(0.05\) is at \(z=1.64\)

Example: The Crash - Model Validation

We assumed returns follow normal distribution. Using an inaccurate model can lead to inaccurate results.

S&P 500 Returns vs Normal

Normal as Approximation to Binomial

A real estate firm offers a free trip. Of those who accept, 5% buy a property. If the firm brings \(n=1000\) people, what is \(P(X \ge 60)\) for \(X\sim\mathrm{Binomial}(1000, 0.05)\)?

Mean \(\mu=np=50\), \(\sigma=\sqrt{np(1-p)}=\sqrt{47.5}=6.89\). Continuity correction: \[Z=\frac{59.5-50}{6.89}=1.38, \qquad P(Z\ge 1.38)\approx 0.084\]

In R the exact value is a one-liner, and the approximation is there to be checked, not to replace it:

1 - pbinom(59, 1000, 0.05)   # 0.087
1 - pnorm(59.5, 50, sqrt(47.5))