In general, the

*theory of decision-making under uncertainty*suggests maximizing an expected value, such as*expected profit*(theory of the firm),*expected utility*(theory of the consumer), or*likelihood*(maximum likelihood estimation in statistics).In betting where you either win a prize or you don't, if two events have the same value prize, we can simplify to betting on the highest probability.

Here we have events A, A', and B where:

- A = urn contains two red marbles and one blue marble
- A' = urn contains two blue marbles and one red marble
- B = student draws red marble

The student must bet on

*more red marbles urn*("red") or*more blue marbles urn*("blue"). He bets on "red" when he sees red if P[A|B] > 1/2. (Since the urn is known to be either "red" or "blue", in that case P[A'|B] < 1/2, so "red" has higher probability.)