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10 · Betting games

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Probability, lesson 10

Betting games

Fair games, edges, and how much to bet when the odds favour you.

The last lesson showed that a real edge pulls clear of the noise, given enough bets. This one is about collecting it: what makes a bet fair, what each bet does to your money, and how much of it to stake. You need expected value: each outcome times its chance, added up.

By the end, you can:

  1. Measure a bet's edge, and find the chance that makes it fair.
  2. Work out what a run of wins and losses does to your money.
  3. Find the share of your money that grows it fastest.

Part 1. Fair games and edges

Measure a bet's edge, and find the chance that makes it fair.

Odds are usually given as a payout. A bet at 3 to 1 pays 3 for every 1 you stake if it wins, and takes your 1 if it loses. Whether that's a good bet depends on your chance of winning. The payout tells you the chance at which it stops being one.

Example 1.1

A bet at 3 to 1

A bet pays 3 to 1. What chance of winning makes it fair? And if you win it 30% of the time, how much do you expect to make for each 1 you stake?

  1. Write the expected profitp × 3 − (1 − p) × 1

    Win 3 with chance p, and lose your stake of 1 otherwise.

  2. Set it to zero4p − 1 = 0, so p = 1/4

    At a 25% chance of winning, neither side has an edge.

  3. Put in your chance0.3 × 3 − 0.7 × 1 = 0.2

    At 30%, each 1 you stake earns 0.2 on average.

AnswerFair at a 25% chance of winning. At 30%, you expect 0.2 for every 1 you stake.

Odds of b to 1 are fair when your chance of winning is 1 / (b + 1). A win pays b and a loss costs 1, so to break even you need one win for every b losses. At 3 to 1 that's 1 win in 4. Each point above 25% is profit, on average.

Fair game and edge

A fair game has an expected profit of zero for both sides. Your edge is your expected profit for each 1 you stake. For a bet that pays b to 1 and wins with chance p:

edge = p × b − (1 − p)

No way of sizing bets changes the sign of an edge. A well-known system doubles the stake after every loss on a fair coin, so the first win recovers every loss plus 1. With 1,023 chips you can stake 1, 2, 4 and so on up to 512: ten bets. You win 1 unless all ten lose.

All ten lose with chance 1/1,024, and that costs all 1,023 chips. The average is 1,023/1,024 × 1 − 1/1,024 × 1,023 = 0. The system swaps many small wins for one rare, huge loss. Every bet is fair, so their total is fair too.

Stakes when you double after every loss 1 2 4 8 16 32 64 128 256 512 bet 1 bet 10 The first win, at any bet: +1 All ten lose, 1 time in 1,024: −1,023
Stakes when you double after every loss 1 2 4 8 16 32 64 128 256 512 bet 1 bet 10 The first win, at any bet: +1 All ten lose (1 in 1,024): −1,023
Figure 1Doubling after every loss. Ten stakes, from 1 to 512, use 1,023 chips. The first win, at any bet, nets 1. Ten losses in a row, 1 time in 1,024, lose them all.

Your turn

A bet pays 4 to 1, and you win it with chance 1/4. What's your edge for each 1 you stake?

0.25 × 4 − 0.75 × 1 = 0.25. Odds of 4 to 1 are fair at a chance of 1/5, not 1/4. The answer 1 forgets the stake you lose three times in four.

Part 2. Bets multiply your money

Work out what a run of wins and losses does to your money.

Keep going with Pro

You've done the first part. Pro opens the rest of this lesson, and every other one.

Still to come
  1. Bets multiply your money
  2. The Kelly bet
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Next lessonRandom walksGambler's ruin, hitting times, and how far a random path wanders. Start lesson 11

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