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Winning the Lottery

Age 7 to 11 Challenge Level:

Winning the Lottery

In a far-away land, the lottery consists of four balls numbered $1$ to $4$, which are placed in a bag.
To enter, you choose one number.
To win, your number must match the number that is drawn from the bag.

What is the chance of winning this lottery?
The people running the lottery in this far-away land decide that it is too easy to win.  So, they change their lottery game.
In the new lottery, there are still four balls numbered $1$ to $4$, which are placed in a bag.
Now, to enter, you choose two numbers.
To win, your numbers must match (in any order) the two numbers that are drawn from the bag.
What is the chance of winning this new lottery?
Have the organisers made it harder to win compared with their original version?
Can you create your own version of the lottery which would also be harder to win than the first game? 
How do you know that your game is harder?
You may like to use our lottery simulator to try out your ideas:

Why do this problem?

This problem offers an engaging context for developing children's understanding of experimental and theoretical probability.  The interactivity will give pupils a 'feel' of the situation. Through the experimental data, then they can be encouraged to calculate theoretical probabilities by working systematicallly, and listing all possible winning combinations.

Possible approach

Ideally, it would be good to have access to the Lottery Simulator interactivity as you introduce and work on this problem.  If this is not possible, you could simulate the lottery yourself by having numbered balls or digit cards in a bag.  You could pick out the 'winning' number yourself and record the results somewhere for all to see.   
To begin with, you could set up the interactivity so that only four balls are available in the 'Number Tumbler' and 'your ticket' has just one number.  Explain the way this lottery works to the group and invite suggestions for the number to choose on your ticket.    You could click on the subtraction sign in the 'Lottery Simulator' panel to select just one draw and then click the 'Simulate lottery draws' button to see what happens.    You could simulate a draw a few times.
Once the group has got a feel for what the interactivity is doing, you could pose a few questions, such as:
What is the chance of winning our lottery?
How many times would you expect to win in ten draws?
How many times would you expect to win in twenty draws?
How many times would you expect to win in fifty draws? 
How many times would you expect to win in one hundred draws?
Give children time to work on these questions in pairs, making it clear that you will be expecting them to explain their thinking.
After a suitable period, bring the whole class together to share their ideas and to ask them to  justify their predictions mathematically.  For example, some children might  say that there are four possible winning numbers ($1$, $2$, $3$ or $4$) and if you have chosen one of them on your ticket, the chance of you winning is $1$ out of $4$ or a quarter or $25$%. 
You could use the interactivity to simulate ten/twenty/fifty etc draws.    Did the simulations match their predictions?  This is a good chance to bring up the idea that the higher the number of draws, the more likely the experimental data (i.e. the simulator) will match the theoretical data (i.e. one in four).
You can encourage the group to investigate the modified version of the lottery in a similar way.  This time, it is more of a challenge to list all the possible draws.

Key questions

What number/s could be drawn?
What are all the possible draws? 
How do you know you have got them all?
How can you tell which version of the lottery is easier/harder to win?

Possible extension

Encouraging learners to find their own ways to make it more difficult to win the lottery is a never-ending task!  You may be surprised to observe those who stick to situations which they feel comfortable analysing and those who push themselves a bit further. 

Possible support

It might be helpful for learners to have access to the interactivity in pairs.