Chances are

Which of these games would you play to give yourself the best possible chance of winning a prize?

Age

14 to 16

Challenge level

Secondary

STATISTICS AND PROBABILITY

Probability

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

Chances Are printable sheet

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.

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 $\frac{1}{10000}$

Christian from Takeley explains how to calculate the probability for number 3:

The game which involves the ten famous pictures. You work out the probability of putting the right picture first, which is one tenth. You then work out the probability of getting the right picture and putting it into the second place, which is one ninth. You continue in this way until you have all four individual probabilities. You then multiply them all together: one tenth, one ninth, one eight and one seventh, to get your final answer. The probability is $\frac{1}{5040}$, which by coincidence is the same probability as in the gardener game.

Well done too to Patrick from Woodbridge School and Yash from Natomas Charter School who sent in largely complete solutions to this problem. Yash commented:

Since all the numerators are the same, the greatest probability would be have to be the least denominator. Thus you'd be best off tossing a coin 12 times, but it's probably not as much fun.

And Patrick pointed out:

Thus, the best deal seems to be the twelve heads in a row - which is so unlikely that it would be better to buy that bottle of water instead!

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.

Now give the groups time to write down the calculation they would use to represent each situation, encouraging use of fractions rather than decimals for easier working without calculators.

Check that groups have modelled each situation appropriately. This is a good opportunity for some discussion about the more subjective options - those which involve a ranking based on preference - and how the probability would change given more information about the people who put them in order.

Once everyone has a model for each option, groups can start to compare different options. Some comparisons are easier than others, so encourage use of prime factorisation to compare the size of the denominators. Laws of indices can be useful to compare for example $(\frac{1}{2})^{12}$ with $(\frac{1}{10})^4$.

Once comparisons have been made, students should compare their answers with their intuition about which was best, and consider whether they would try to win the holiday or keep their money! Another discussion point could be how much the prize should be worth in order for the organisers to make money from the enterprise.

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.

The Better Bet is another problem about choosing which game to play in order to maximise winnings.

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