152→ Odds

Expected value

Expected value is what a bet wins or loses on average, if you repeated it many times. Enter the odds and the probability: the calculator does the rest.

Expected value per bet-$55-5.5% of the stake, on average

One bet never pays the average: you win $1,100 or lose $1,000. The average only shows up over many bets.

You need at least47.6%probability to break even
Fair odds for you2.221 divided by your probability
Spread of one bet$1,045standard deviation

100 bets like this one

$0100 bets

middle result after each betwhere 8 in 10 runs fall$0: neither up nor down

Left to right: from the first bet to bet 100. Up: money ahead; down: money behind.

Ahead after 10030.7%
Behind after 10069.3%
Average result-$5,500of $100,000 bet

8 in 10 runs of 100 bets end between -$18,100 and +$7,100.

Show the formula

Expected value = p × (odds − 1) × stake − (1 − p) × stake

45.0% × $1,100 − 55.0% × $1,000 = -$55.00

Break-even probability = 1 ÷ odds

Spread = stake × odds × √(p × (1 − p))

Illustration: Expected value

The formula

p × profit − (1 − p) × stake

Example: at odds 2.10 and 45%: 45% × $1,100 − 55% × $1,000 = -$55 per bet.

How to read it

  1. Two pathsIf it wins, you make $1,100. If it loses, you lose $1,000.
  2. Weigh each oneMultiply each path by its probability and add them. That is the expected value.
  3. Break-evenAt odds 2.10 you need at least 47.6% probability not to lose on average.
  4. Many timesOver 100 bets like this, you end ahead only 30.7% of the time and lose $5,500 on average.

Same odds, different probabilities

Your probabilityExpected value
40%-$160
45%-$55
47.6%$0
50%+$50
55%+$155

A $1,000 bet at odds 2.10. A few points of probability flip the sign.

Your estimate or the market's

My estimate

  • You set the probability
  • Shows what you would have to believe
  • Easy to fool yourself

Market without margin

  • Comes from the 1X2 odds
  • With 2.10 · 3.30 · 3.60 it gives -$54 per bet
  • That is the margin, 5.4%, against you
Bookmaker marginWhere the negative expected value of almost every bet comes from.A thousand parlaysThe average at work: a bankroll that plays many times.

Questions

Does a positive expected value mean I will win?

No. It means that if your probability were right, the average over many bets would be positive. Each single bet is won or lost in full.

Why is it always negative with the market probability?

Because the price has the bookmaker margin built in. With the proportional method every outcome loses the same share on average: 5.4% in the example.

What is the spread?

How far, typically, the result of one bet lands from its average. With long odds it is huge: the average takes a long time to show up.