Chances are
Which of these games would you play to give yourself the best possible chance of winning a prize?
Problem
Here are five competitions you could enter.
Which one offers you the best chance of winning?
- You win a prize if you flip a fair coin and get twelve heads in a row.
- Our gardener has ranked her seven favourite plants in order. If you rank them in the same order, you win.
- Choose the top 4 from 10 famous pictures and put them in the right order to win.
- Throw five fair dice and get five sixes to win the first prize.
- You throw four ten-sided dice and win first prize if you get four sixes.
Click here for a poster of this problem.
Getting Started
The answers are not necessarily all different!
Student Solutions
Rajeev from Fair Field School sent us the following solution:
I found that your best chance of winning was the option of tossing the coin and the chance of winning was $\frac{1}{4096}$.
Here is how I worked it out:
- You win first prize if you can toss a fair coin and get 12 heads in a row. With one coin toss, you get half a chance, with 2 coin tosses you get $\frac{1}{4}$ and with 3 you get $\frac{1}{8}$ so with 12 coin tosses you get$(\frac{1}{2})^{12}$ which is $\frac{1}{4096}$
- Throw 5 fair dice and you get 5 sixes and you win the first prize. With 1 fair die your chance of getting a six is $\frac{1}{6}$ and with 2 its $\frac{1}{36}$ so with 5 fair dice its $(\frac{1}{6})^5$ which is $\frac{1}{7776}$
- Choose the top 4 from 10 famous pictures and put them in the right order to win. $\frac{1}{5040}$
- Our resident Gardener has listed her seven favourite plants in order. If you can match the order you win. With 2 there are 2 ways of ordering, with 3 it is 6 and with 4 it is 24 and so with 7 it is $7\times6\times5\times4\times3\times2\times1)=5040$ So the probability of selecting the correct ordering is $\frac{1}{5040}$
- You toss four ten-sided dice and win the first prize if you can get 4 sixes. With one die it's $\frac{1}{10}$ and with 2 dice it's $\frac{1}{100}$ and with 4 dice it is $\frac{1}{10000}$
Teachers' Resources
Why do this problem?
This problem offers a great opportunity for talking about chance. Students can use their intuition to rank the options in order, and then model the situation and manipulate the resulting fractions. Finally, there is a chance to discuss whether the models used are appropriate and how this might affect their answer.
Possible approach
Present the problem, and explain that students are being asked to try their luck without having access to anything more sophisticated than paper and pencil. Allow some time for pairs or small groups to discuss the different options, making sure they understand what they mean. Each group should come up with which option they think gives them the best chance or the worst chance of winning, with some justification for why they believe it.
Key questions
Which game do you think is easiest to win?
How can we compare two fractions with different denominators?
Possible support
Allow some calculator use so the focus is on the probability calculations without the comparisons of fractions getting in the way.
Possible extension
Come up with other scenarios with similar probabilities.