Expected value

Expected value answers one question: if a bet could be repeated indefinitely under identical conditions, what would the average result per trial settle at? It is computed by multiplying each possible outcome by the probability of that outcome and adding the products. Nothing else enters the calculation, and in particular no prior result does.

EV  =  (p1 x v1) + (p2 x v2) + ... + (pn x vn)

  where p is the probability of an outcome
  and   v is the amount won or lost on it

A worked case

Take one unit staked on a single number of a thirty-seven-pocket wheel, paid at 35 to 1. There are two outcomes. The number appears, with probability 1/37, returning a profit of 35 units. It does not appear, with probability 36/37, losing the one unit staked.

  (1/37  x  +35)  +  (36/37  x  -1)

= (35/37)          +  (-36/37)

= -1/37  =  -0.027027 units   per unit staked

Two features of that result are worth stating plainly. The first is that no single trial ever produces −0.027 units; every trial produces either +35 or −1. Expected value is a property of the distribution, not a prediction about any trial. The second is that the figure is negative, and it is negative because the prize paid is 35 rather than the 36 that would make the bet break even.

Adding trials

Expected values add. Staking one unit on each of n independent trials gives an expected result of n × (−1/37), so expected loss grows in a straight line with the number of trials: 2.70 units over a hundred trials, 270 units over ten thousand. This is the sense in which a negative expectation is not a risk that may or may not materialise. It is the centre of the distribution, and the distribution narrows around that centre as trials accumulate.

A fair game

A game whose expected value is zero is described as fair. Fairness in this sense is a statement about prices, not about honesty: a game can be scrupulously honest, with a perfectly uniform wheel and a fully certified selection mechanism, and still carry a negative expected value, because the edge lives in the prize schedule rather than in the mechanism. No commercially offered game has an expected value of zero or above, because a game without an edge would produce no revenue.