### 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?

### Why do this problem?

This problem provides a motivation to use sample space diagrams; it could be used to introduce or consolidate work on them. The interactivity offers an ideal context in which to observe the "messy" randomness of results after a small number of experiments, and the predictability of results after a large number of trials.

The problem also offers a good starting point for considering different probability distributions and their features.

### Possible approach

Interactive Spinners provides a useful introduction to this problem, and the Teachers' Notes suggest how it could be used in the classroom.

Introduce the interactivity with a single spinner, one spin at a time. Clarify that it is not a frequency table, it's a relative frequency table.

"What could happen to the chart after the next spin?" Collect together the different possibilities together with the students' explanations.
"Let's see which one happens" and spin again.
Repeat until students are secure in their understanding of the relationship between what is happening on the spinner and what appears on the relative frequency chart.
Finally, "What would the chart look like after $50 000$ spins?" Allow time for them to discuss in pairs, then gather together suggestions and justifications before spinning to confirm their ideas.

If computers are available, ask students to work in pairs and set them the following challenge:

"I wonder whether you could predict what would happen if we had two spinners...
In a while, I'm going to choose two spinners (which could be identical but may be different), and decide on either the sum or the difference, and you will need to be able to predict what the bar chart will look like, and explain how you know."

"You have some time to work in pairs at the computer to prepare for this challenge by experimenting with different pairs of spinners and recording what you notice in order to help you to make predictions." Show how to set up the interactivity with two spinners, if necessary.

While they are working, circulate and listen to students' noticings. Challenge them to explain what they have noticed, and be aware of any students who have useful recording methods, insights or explanations that could be shared with the rest of the class.

Bring the class together to share their insights and explanations before handing out this worksheet.
"Can you work out how these graphs were created, WITHOUT using the interactivity?"

Once pairs have had time to decide how the eight graphs were created, a nice way to finish off the task is to arrange the class into eight groups, give each group one of the graphs, and invite them to prepare a short explanation to present to the class.

If computers for the students are not available...

"What would happen if we had more than one spinner? I'm going to set the interactivity up with two spinners going from 1 to 3, and each time, it will record the sum of the two numbers. With your partner, talk about what you think the graph might look like after $50 000$ spins."

Once they've had a chance to discuss, share ideas and ask for justifications before checking using the interactivity. If no-one has suggested using a sample space diagram, this would be a good opportunity to introduce the technique to explain the heights of the bars on the chart.

Hand out the first page of this worksheet. "These graphs were created using two spinners. Your challenge is to work out which two spinners were used in each case, and provide a convincing argument."

Give students plenty of time to work on the challenge in pairs. As they are working, circulate and note which pairs have insights that are worth sharing.

The second page of the worksheet could be handed out to pairs who have identified the spinners for the first page, for them to start thinking about.

Bring the class together to discuss the first page, and invite those pairs with interesting ideas to share what they did, using the interactivity to check. Then hand out the second page to all and set them the same challenge as before. "These graphs look rather different - see what key features you notice and see if you can deduce how they were made and which spinners were used."

Again, give students time to work on the challenge before bringing the class together to share what they found. If time allows, the first extension task below would provide a good final challenge.

Here is a version of the interactivity which allows you to choose the numbers on the spinners, rather than being restricted to consecutive numbers starting at $1$.

### Key questions

What features of the chart do you think might be important?
How could those features have been created?
Can you deduce anything about the biggest numbers on the spinners?

### Possible extension

Set students the final challenge from the problem:

"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 me that your descriptions of the bar charts are correct."

A more challenging extension is to explore the charts produced by three spinners.

### Possible support

Interactive Spinners provides a good introduction to this problem.

Begin by just exploring the first four graphs where only the sum is used rather than the difference. Then introduce the second page of the worksheet with an example of two 1 to 3 spinners finding the difference rather than the sum, and take time to show how a sample space diagram can explain the features of the graph.