probability, odds, expectation

Counting the outcome set

A probability is a count divided by a count. Before any figure can be quoted, the set of outcomes has to be enumerated and each member has to be equally likely — or, if it is not, broken down further until it is. A single die has six outcomes. Two dice do not have eleven outcomes, they have thirty-six: the sums run from two to twelve, but the sums are not equally likely, and treating them as if they were is the oldest error in the subject.

This is the whole discipline. Where a figure on this site looks surprising, the surprise is almost always in the outcome set rather than in the arithmetic that follows it. A wheel with thirty-seven pockets and a payout table written for thirty-six is the entire explanation of roulette, and it takes one line to state.

Three ways of writing the same ratio

Odds and probability carry identical information. Probability is a share of the whole; odds are a ratio of the two parts. Fractional notation quotes the profit against the stake, so 5/2 means two units risked to win five. Decimal notation quotes the total return per unit staked including the stake, so the same price is 3.50. Moneyline notation quotes what a hundred units wins, or what must be risked to win a hundred, so the same price is +250.

Converting to implied probability is the only operation that matters, and decimal makes it trivial: the implied probability is one divided by the decimal price.

Expected value

Expected value is the sum of every outcome's value multiplied by its probability. It answers a narrow question — what does this wager average, repeated without end — and it answers nothing about the next round. That narrowness is exactly why it is useful. It is a fact about the rule set, not a forecast, and it can be computed before the game is ever played.

A wager whose expected value is zero is called fair. Commercial games are not fair in this technical sense, and are not meant to be: the gap between the true odds of an event and the odds paid on it is the operator's revenue, and it is the subject of the next entry.

E[X] = sum over outcomes of ( value of outcome x probability of outcome )

example: a bet paying +5 with probability 1/8, losing 1 otherwise

E[X] = (+5)(1/8) + (-1)(7/8)
     = 0.625 - 0.875
     = -0.25 per unit staked

the true price of a 1/8 shot is 7/1; paying 5/1 costs the bettor 25% of stake

Where the notation misleads

The second recurring trap is arithmetic on percentages rather than on units. A game that returns 97% of stake does not return 97% of a bankroll, because a bankroll is staked many times over. That distinction is the whole content of the long-run entry, and it is where most intuition about these games quietly fails.