Random variables and expected value

A random variable turns each outcome into a number: the face of a die, the money a game pays, the number of heads.

Expected value

The expected value E[X] is the average you would get over very many repeats:

E[X] = sum of (value × its probability)

For a fair die: (1 + 2 + 3 + 4 + 5 + 6) × 1/6 = 3.5. You can never roll 3.5; it is a long-run average.

Variance and standard deviation

Variance is the expected squared distance from the mean: Var(X) = E[(X − E[X])²]. Its square root, the standard deviation, is in the same units as X and says how far values typically land from the mean.

faces = [1, 2, 3, 4, 5, 6]
mean = sum(faces) / 6
variance = sum((x - mean) ** 2 for x in faces) / 6
print("E[die] =", mean)
print("Var =", round(variance, 3), " SD =", round(variance ** 0.5, 3))

payout = 10 * 0.1 + 0 * 0.9
print("ticket costs 2, expected payout", payout, "so expected gain", payout - 2)

Is the game worth it?

A ticket costs 2 and pays 10 with probability 0.1, else nothing. E[payout] = 10 × 0.1 = 1, so on average you lose 1 per ticket. Every casino game has a negative expected value for the player.

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