### Win or Lose?

A gambler bets half the money in his pocket on the toss of a coin, winning an equal amount for a head and losing his money if the result is a tail. After 2n plays he has won exactly n times. Has he more money than he started with?

### 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?

### Scratch Cards

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?

# Coin Lines

### Why do this problem:

Probability often forces us to be particularly careful with our justification of answers. This problem has a simple enough numerical answer but the visualisation to support it must be carefully considered.

### Possible approach :

This problem might make a good poster, displayed somewhere it will catch students' attention to promote discussion.

The extent to which students need some practical activity will depend on how accustomed they are with visualisation tasks.

### Key questions :

• What do you think the answer might be?
• Do you have a way of looking at this situation so that you are sure your answer is right?

### Possible extension :

• Research the problem context called Buffon's Needle

### Possible support :

For students who cannot access this problem directly or theoretically the following activities may be helpful :
• Draw some parallel lines at equal intervals and vary the size of that interval. Include in particular double and treble the coin diameter. Keep a tally of results.
• Draw the concentric circles and collect experimental data.