courses › Beyond College

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

Pen and paper is fine · no calculator needed why?

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Opens at level 59.

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Builds on: Expected value & variance (not open yet) · Eigenvalues (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Long-run share par 60 s not tried yet

  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 par 30 s not tried yet

  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 par 30 s not tried yet

  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 par 30 s not tried yet

  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.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

your rounds

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