Probability distributions, expectation and fair games

Last time

Before we start: last time, in Selections without replacement, tree diagrams and conditional probability, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

What to expect

A game of chance may win or lose money on any one turn, but over many turns its average result is predictable. This lesson covers probability distributions and expectation, expected profit and loss, and fair games.

Probability distributions

A probability distribution lists every possible value of a variable, with its probability. The probabilities must add up to 1. For the number of heads when two coins are tossed: 0 with probability a quarter, 1 with a half, and 2 with a quarter.

Expectation

The expected value is the sum of each value times its probability. For the two coins: 0 times a quarter, plus 1 times a half, plus 2 times a quarter, which is 1 head. It is the average over many trials, and it need not be a value that can actually occur.

Finding a missing probability

If the probabilities are 0.2, 0.3, p and 0.1, they must add up to 1. So p is 1 minus 0.6, which is 0.4. Always check the total before finding an expectation.

Expected winnings

A game costs 2 dollars a turn. A player wins 5 dollars with probability 0.2, 1 dollar with probability 0.3, and nothing otherwise. The expected winnings are 5 times 0.2, plus 1 times 0.3, which is 1 dollar 30. The player's expected profit is 1.30 minus 2: a loss of 70 cents a turn.

The game as a distribution

The game's winnings as a probability distribution.

The organiser

The organiser gains what the player loses: 70 cents a turn, on average. Over 100 turns, the organiser expects a profit of 100 times 70 cents: 70 dollars. The actual profit will vary; the expectation is the long-run average.

Fair games

A game is fair when the expected profit is zero: the cost of a turn equals the expected winnings. So this game would be fair at a cost of 1 dollar 30. To make a game fair by changing a prize, write the expected winnings with the prize unknown, set them equal to the cost, and solve.

Mistakes that lose marks

Mistake one: forgetting to subtract the cost of playing when finding the expected profit. Mistake two: probabilities that do not add up to 1. Mistake three: confusing the player's profit with the organiser's. The fix: a table, a check of the total, and the cost.

The full-marks tip

The full-marks tip: set out the distribution as a table, check the probabilities add up to 1, then find the expectation. Say whose expected profit you found, per turn, and whether the game is fair.

On one card

So, on one card: probability distributions and expectation. Keep it for revision.