### 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?

# Which Spinners?

##### Stage: 3 and 4 Challenge Level:

We have two spinners, each numbered 1 to 4.
We're going to spin them both and add the numbers.
What possible totals might you get?
What is the least likely/most likely total to occur?

Use the interactivity below to test your hypotheses.

Full Screen Version (click 'back' on your internet browser to exit full screen mode)

This text is usually replaced by the Flash movie.

Now make a new set of spinners.

The interactivity can record either the sum of, or the difference between, the numbers on the spinners, and you can see the results displayed on the relative frequency bar chart.

You can choose to run the interactivity lots of times so that the bar chart "settles down".

Experiment with different pairs of spinners.

What features do you notice on the bar charts that you produce?

Can you come up with ways of predicting what a chart will look like before you produce it?

The challenge

The bar charts below were generated on the interactivity using different combinations of spinners. (You can download a pdf with all eight bar charts here.)

 A B C D E F G H

Can you deduce which spinners were used to create each bar chart?
Can you explain how you used the information provided by the bar charts to work it out?

Final challenge

Imagine you had 1-20 and 1-30 spinners. Describe in as much detail as you can what the relative frequency bar charts would look like for:

• The sum of two 1-30 spinners
• The difference between two 1-20 spinners
• The sum of a 1-20 and a 1-30 spinner
• The difference between a 1-20 and a 1-30 spinner

Try to provide a good explanation to convince us that your descriptions of the bar charts are correct.