The Birthday Bet

Why do this problem?

This problem deliberately involves the consequences of an event (the prize size) not just the chance of the event occurring.

Possible approach

Progress with the problem requires stage-by-stage thinking - and also may require starting with smaller numbers (eg. three people) and then building up from there.

Key questions

• What is the chance that one person's birthday is different from that of another person chosen at random ?

• So what is the chance that their birthdays do match ?

• Given that the first two did not match, what is the chance that a third person will not match with either of those first two ?

• Can you now calculate the probability that all three birthdays are different ?

Possible extension :

The concept of gambler's ruin is useful to include, where the winning system cannot be continued because a series of losses has caused the situation where there is nothing left to bet with.

However, lots of things are not certain and we often need to make decisions in the face of that uncertainty.
Probability is how mathematicians quantify uncertainty.'

Possible support :

Explore the situation where two people each roll a dice and ask how likely is it that they roll the same number. Consider the complementary situation where the numbers rolled must be different. Extend that to three people and onwards, asking at each stage what prize for a £1 stake would be a good-odds bet.