How do you decide if a game is fair using theoretical or experimental probability?

Answer 1

Suppose that each time the game is played you win an amount X, with probability p or lose an amount Y with probability (1-p). To start with it is assumed that you cannot draw a game. Then each time the game is played you expect to win X with a probability p and lose Y (= win -Y) with probability (1-p).

Therefore, the expected gain for you, each time the game is played, is

X*p - Y*(1-p).

The game is fair if this value is 0. If it is greater than 0 the game is biased in your favour. Otherwise it is biased against you.

You can easily incorporate multiple outcomes - such as with a lottery: a bigger win for getting more numbers.

If X(1), X(2), ... X(n) are the n pay-outs to you (remember that what you pay will be a negative number), and P(1), p(2), ... p(n) are their respective probabilities where the sum of all the p(r) = 1, then your expected win per game is

X(1)*p(1) + X(2)*p(2) + ... + X(n)*p(n)

and the game is fair if this is 0.