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Probability theory

lesson · about 3 minutes

Long-run behaviour, random walks, and clever expectations.

Pen and paper is fine · no calculator needed why?

Opens at level 59. You're level 1. You can read and practice here now.

the idea

Each problem here has a short exact answer. The tools are balance, fairness, symmetry and adding up waits.

Symmetry: n independent uniform random points dropped on the line from 0 to 1 cut it into n + 1 gaps of equal average size. So the largest point averages n/(n + 1), and the smallest 1/(n + 1).

techniques

Balance the flows

Two states, with fixed chances of switching each step.

  1. Let p be the long-run share of time in A, so 1 − p is the share in B.
  2. Balance: p × (chance A to B) equals (1 − p) × (chance B to A). Solve for p.
worked example

Example: A process has states A and B. Each step it moves A → B with probability 0.2 and B → A with probability 0.6, and otherwise stays put. What long-run fraction of the time is it in A? Give an exact fraction.

  1. 0.2p equals 0.6 × (1 − p), so 0.8p equals 0.6.
  2. p = 0.6 ÷ 0.8 = 3/4.

Answer: 3/4

A fair game keeps the average

Betting $1 at a time until you reach a target or go broke.

  1. Fair game: you end with the target N or nothing, so N × (chance of reaching N) equals your start.
  2. Uneven odds, win chance p, loss chance q, start i: the chance is ((q/p)ⁱ − 1) ÷ ((q/p)ᴺ − 1).
worked example

Example: You bet $1 at a time on fair coin flips, starting with $4, and stop at $10 or at $0. What is the probability you reach $10? Give an exact fraction.

  1. Your expected money stays $4, so 10 × P equals 4.
  2. P = 4/10 = 2/5.

Answer: 2/5

Add up the waits

Expected tries to see every one of n equally likely types.

  1. With k types missing, each try finds a new one with chance k/n: n/k tries on average.
  2. Add the waits from the number missing now down to 1, then round.
worked example

Example: A spinner has 4 equal sections. How many spins do you expect to need before it has landed on every section at least once? Round to 1 decimal place.

  1. Waits with 4, 3, 2, then 1 missing: 4/4 + 4/3 + 4/2 + 4/1.
  2. That is 25/3 ≈ 8.33, which rounds to 8.3.

Answer: 8.3

watch out for

practice

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Long-run share

worked example

A process has two states, A and B. Each step, from A it moves to B with probability 0.7 (otherwise it stays in A), and from B it moves to A with probability 0.4 (otherwise it stays in B). In the long run, what fraction of the time is it in B? Give an exact fraction.

Answer: 7/11

  1. Let p be the long-run share of time in B. In balance, the flow B → A equals the flow A → B: 0.4p = 0.7(1 − p).
  2. So (0.4 + 0.7)p = 0.7, and p = 0.7 ÷ 1.1 = 7/11.

Gambler’s ruin

worked example

You bet $1 at a time on fair coin flips, starting with $1. You stop when you reach $15 or go broke. What is the probability you reach $15? Give an exact fraction.

Answer: 1/15

  1. In a fair game your expected money stays $1.
  2. If p is the chance of reaching $15: 15p + 0 × (1 − p) = 1, so p = 1/15.

Expected maximum

worked example

You draw 8 independent random numbers uniformly between 0 and 1. What is the expected value of the smallest? Give an exact fraction.

Answer: 1/9

  1. The 8 draws split [0, 1] into 9 gaps of equal expected size, 1/9.
  2. The smallest value is the first gap: 1/9.

Collect them all

worked example

Each cereal box holds one of 2 different toys, all equally likely. How many boxes do you expect to open before you have all 2 toys? Round to 1 decimal place.

Answer: 3

  1. When k of the 2 toys have appeared, a new one comes with chance (2 − k)/2, so it takes 2/(2 − k) tries on average.
  2. Total: 2 × (1 + 1/2) = 2 × 3/2 = 3.

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