### Coin Tossing Games

You and I play a game involving successive throws of a fair coin. Suppose I pick HH and you pick TH. The coin is thrown repeatedly until we see either two heads in a row (I win) or a tail followed by a head (you win). What is the probability that you win?

### Fixing the Odds

You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two bags so as to make the probability of choosing a red ball as small as possible and what will the probability be in that case?

### Penta Colour

In how many different ways can I colour the five edges of a pentagon red, blue and green so that no two adjacent edges are the same colour?

# 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 :

This problem allows a wider exploration and research of 'systems' to win, at Roulette for example.

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.

There is also lots to discuss in the end note'
No matter how big the prize or how easy it looks to win, it isn't smart to bet if I can't stand the loss.
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.