courses › Thinking

How much to bet, how to mix strategies, and what information is worth.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 49.

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Builds on: Risk & the long run (not open yet) · Exponentials & logarithms (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.

• Information in bits par 20 s not tried yet

  worked example

  Four outcomes have probabilities 1/2, 1/4, 1/8 and 1/8. What is the entropy in bits?

  Answer: 1.75 bits

  1. An outcome with chance 1/2ᵏ carries k bits. Entropy = Σ p × bits.
  2. 1/2 × 1 + 1/4 × 2 + 2 × 1/8 × 3 = 0.5 + 0.5 + 0.75 = 1.75 bits.

• How much to bet par 30 s not tried yet

  worked example

  A bet pays 4-to-1 (win $4 for each $1 staked; lose the $1 otherwise), and you win 46% of the time. What fraction of your bankroll does the Kelly rule say to stake? Give a percent.

  Answer: 32.5%

  1. Kelly: f = p − q/b = 0.46 − 0.54/4 = 0.325, so stake 32.5%.
  2. (Many people bet half of that to cut the swings.)

• Mixed strategies par 45 s not tried yet

  worked example

  A zero-sum game: you pick High or Low, and so does your opponent. Your payoffs: you High, they High: +4; you High, they Low: −1; you Low, they High: +3; you Low, they Low: +4. Your opponent gets the negative of your payoff. At equilibrium, with what probability should you play High? Give a fraction.

  Answer: 1/6

  1. Choose p (your chance of High) so your opponent can't exploit you.
  2. If they play High, you average 4p + 3(1 − p) = p + 3; if Low, −p + 4(1 − p) = −5p + 4.
  3. Setting them equal: 6p = 1, so p = 1/6.

• What information is worth par 45 s not tried yet

  worked example

  Opening a second store earns +$130,000 if foot traffic is strong (20% chance) and −$460,000 if it's weak. A perfect traffic study would tell you which before you decide. What is the most you should pay for the study?

  Answer: $26,000.00

  1. Without it, opening expects 0.2 × $130,000 − 0.8 × $460,000 = −$342,000, so you wouldn't: $0.
  2. With it you go ahead only when traffic is strong: 0.2 × $130,000 = $26,000.
  3. The information is worth $26,000 − $0 = $26,000.

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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