Probability and expected value
Sample spaces, probability as a proportion of equally likely outcomes, the two combining rules, and expected value as a probability-weighted average.
Counting the outcomes
Every probability begins with a list. The list is called the sample space: it is the complete set of results a single trial can produce, written out so that no result is missing and none is counted twice. One die has a sample space of six. Two dice thrown together have a sample space of thirty-six, because each of the six results of the first die can pair with each of the six results of the second. A wheel of thirty-seven pockets has a sample space of thirty-seven.
The reason the list matters more than it first appears is that most mistakes in this subject are mistakes of counting, not of arithmetic. Two dice do not have eleven equally likely outcomes because the totals run from two to twelve; they have thirty-six equally likely outcomes, which happen to produce eleven distinct totals at very different rates. A total of seven arises from six of the thirty-six pairs and a total of twelve from one of them, which is why one is six times as common as the other.
Two dice, 36 equally likely pairs
total pairs probability
2 1 1/36 = 0.0278
3 2 2/36 = 0.0556
4 3 3/36 = 0.0833
5 4 4/36 = 0.1111
6 5 5/36 = 0.1389
7 6 6/36 = 0.1667
8 5 5/36 = 0.1389
9 4 4/36 = 0.1111
10 3 3/36 = 0.0833
11 2 2/36 = 0.0556
12 1 1/36 = 0.0278
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36/36 = 1.0000
Probability as a proportion
Where the outcomes in the sample space are equally likely, the probability of a described result is the count of outcomes that satisfy the description divided by the count of outcomes in total. It is therefore always a number between zero and one, and the probabilities of all the outcomes in a sample space always add to exactly one. That last property is a useful check: a set of probabilities that does not sum to one has either lost an outcome or double-counted one.
Two combining rules cover most cases. If two results cannot both occur on the same trial, the probability that either occurs is the sum of their probabilities. If two results belong to separate, independent trials, the probability that both occur is the product of their probabilities. The two rules are frequently swapped, and the error is easy to spot: adding probabilities across independent trials produces figures greater than one, which is impossible.
Same trial, mutually exclusive: P(total is 7 or 11) = 6/36 + 2/36 = 8/36 = 0.2222 Separate independent trials: P(7 then 7) = 6/36 x 6/36 = 36/1296 = 0.0278
Dependent trials
Independence is an assumption, and it is worth checking rather than assuming. Dice and wheels are independent between trials because nothing about a trial changes the mechanism for the next one. Cards dealt from a shoe without replacement are not: removing a card changes the composition of what remains, so every subsequent probability shifts. The probability that the first card dealt from a full fifty-two card pack is a particular rank is 4/52; the probability that the second is that same rank, given the first was, is 3/51, because both the numerator and the denominator have changed.
From probability to expected value
A probability describes how often. It says nothing about how much. Expected value combines the two by weighting each outcome by its probability and adding the products, which converts a distribution of results into a single average per trial.
A wheel of 37 pockets, one unit on a single number, paid 35 to 1 (1/37 x +35) + (36/37 x -1) = (+0.945946) + (-0.972973) = -0.027027 units per unit staked
The negative sign is not an artefact of this particular bet. It appears because the prize paid, 35 to 1, is smaller than the 36 to 1 that a result occurring once in thirty-seven would have to pay for the average to come out at zero. Any prize schedule set below the break-even schedule produces a negative expected value, and every commercially offered game is set that way, since that gap is the mechanism by which such a game produces revenue.
What the average does and does not tell you
Expected value is an average over repetitions, not a forecast of any single trial. No spin of the wheel above returns −0.027 units; every spin returns +35 or −1. Its usefulness is that averages add cleanly: the expected result of a hundred such trials is a hundred times −0.027027, or −2.70 units, regardless of the order the results arrive in. That additivity is what makes the figure the right tool for comparing games and the wrong tool for describing an evening.