### 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?

# Introducing Distributions

### Why do this problem:

This problem was created as a preliminary problem before students try Data Matching , where it was felt that an introduction to probability distributions would be useful.

### Possible approach :

The aim of this activity is for students to acquire a feel for what a probability distribution is : that it is abstract and though sample data often follows the theoretical distribution more closely the larger the sample size it isn't compelled to do so. It's just a rare event when it doesn't.

Additionally, at Stage 4 students mostly calculate only individual probabilities and it is helpful to also draw attention to the profile of probability across the complete range of values for our variable of interest.

The 'copy to clipboard' facility collects data. Ask students to collect ten sets of data (ten times 100 throws) pasted into Excel.

Keep a data set that has the samples separate and another one which forms a combined sample. These two are to be compared.

Draw the frequency distribution (or relative frequency) distributions for the first 100, the first 200, then 300 and so on.

Encourage students into commenting on the graph for each of these and help them to explain what they observe until this example of randomness has begun to become secure.

### Key questions :

• What does the probability distribution look like for the outcomes when you roll one die, and can you explain why?
• Why isn't the graph on the 'distribution maker' a horizontal line ?

After the samples have been collected and accumulated :

• How many of the samples of 100 are less even than the combined sample of 1000 ?
• Is a larger (or combined) sample always closer to the actual probability distribution ?

### Possible extension :

Abler students may cover the above issues as a class discussion and move quickly on to Data Matching

### Possible support :

For less able students the data acquisition and accumulation is best done as a teacher-led whole group activity so that technical challenges do not obstruct their view of the main question : what is the relationship between the abstract probability distribution and the sample data.