Polson–Sokolov Lab · Streaks, Bernoulli and binomial

Streaks Lab

A one-in-a-million event almost never happens to one person, and it happens all the time to somebody. Count how many chances there were: 32 NFL teams tossing coins, or thousands of hitters chasing a streak over many seasons.

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Based on Chapter 1, Probability and Uncertainty, of Polson and Sokolov, Bayes, Artificial Intelligence, and Deep Learning (vsokolov.org/html/_book).

Simulated team (one dot each) Team in the news region Theory: expected teams, and P(at least one) News region
The law of very large numbers

Same template, different p and different number of chances

The Pete Rose calculation and the Patriots calculation are the same template. Each one multiplies small probabilities for one person. The real question is how many chances there were for the rare event to happen somewhere.

  1. One game is a Bernoulli trial

    Rose hit .300 with 4 at-bats per game. He fails to get a hit only if all 4 at-bats are outs, so P(hit in a game) = 1 − 0.7⁴ = 0.76. For DiMaggio at .325 it is 1 − 0.675⁴ = 0.792. Both use the binomial X ~ Bin(4, p): a hit in the game means X ≥ 1.

  2. A fixed run of games multiplies

    If games are independent, 44 hits in 44 named games has probability 0.76⁴⁴ ≈ 5.7 × 10⁻⁶, about 1 in 175,000. DiMaggio's 56 is 0.792⁵⁶ ≈ 2.1 × 10⁻⁶, about 1 in 456,000. That is the number for one particular stretch of games.

  3. A season has many starting points

    A streak can start in any game. The chance of a 44-game streak somewhere in a 162-game season is about 30 times the fixed-stretch number. The lab gets it exactly with a recursion that tracks the current run length game by game.

  4. Many players and seasons make it likely

    If one season has chance q, then m independent player-seasons give 1 − (1 − q)m. With m ≈ 1/q chances this is already 63%. The Patriots' 19 of 25 is about 1 in 137 for one team, and about 1 in 5 for "some team this season".

Binomial count of toss wins
P(X = k) = C(n, k) pk (1 − p)n−k
P(X = 19) = 177,100 × 2⁻²⁵ ≈ 0.0053
P(X ≥ 19) = 0.0073 ≈ 1 in 137
At least one of m independent chances
P(some team) = 1 − (1 − 0.0073)³² ≈ 0.209
Exact streak recursion. State r = current run length, r = 0, …, L−1.
After a hit: r → r + 1 (prob p)
After a miss: r → 0 (prob 1 − p)
P(streak ≥ L) = total probability of ever reaching r = L
Worked cases (one season, exact)
CasepLSeason
Rose, 0.76, L = 44, 162 g5.7e−61.7e−4
DiMaggio, 0.792, L = 56, 154 g2.2e−64.7e−5

Takeaway for students. Before you are impressed by a rare event, ask how many chances there were for it to happen somewhere, and whether you only heard about it because it was unusual. That is selection bias in the news. It is also why a fund manager with a 10-year winning streak, or a drug trial with one good subgroup, needs the same check.

Bernoulli process

Independent trials with the same p. Two successes in a row have probability p², and L in a row have pL.

P(X₁ = 1, …, XL = 1) = pL

Binomial mean and variance

For 25 fair tosses the mean is 12.5 and the SD is 2.5. Nineteen wins is 2.6 SDs above the mean.

E(X) = np, Var(X) = np(1 − p)

DiMaggio: more than 2 hits

A classic exercise asks for more than 2 hits in 4 at-bats at .325. A solid game is rare even for an elite hitter.

P(X > 2) = 0.093 + 0.011 = 0.104

Try this in class

  1. Start in coin mode. Press "New season" a few times and count how often some dot lands in the shaded news region. Then press "Run 1,000 seasons" and compare with the theory of about 21%.
  2. Set teams to 1. The chance falls to 0.7%. Ask: would the Patriots story feel as surprising if you knew it had a 1-in-5 chance of happening to some team?
  3. Tick "Losing streaks are news too". The chance of some story rises to about 38%, because 6 or fewer wins is just as rare as 19 or more.
  4. Switch to hitting streaks with the Rose preset. Compare pL (1 in 176,000) with the season chance (1 in 6,000). Then raise the number of hitters until the bottom curve passes 50%.
  5. Load DiMaggio and move the batting average from .325 down to .300. The 56-game chance per season drops by about a factor of 9. Small changes in p matter a lot when raised to the 56th power.