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Gambling at Monte Carlo

A man went to Monte Carlo to try and make his fortune. Whilst he was there he had an opportunity to bet on the outcome of rolling dice. He was offered the same odds for each of the following outcomes: At least 1 six with 6 dice. At least 2 sixes with 12 dice. At least 3 sixes with 18 dice.

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Balls and Bags

Two bags contain different numbers of red and blue balls. A ball is removed from one of the bags. The ball is blue. What is the probability that it was removed from bag A?

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Coin Tossing Games

You and I play a game involving successive throws of a fair coin. Suppose I pick HH and you pick TH. The coin is thrown repeatedly until we see either two heads in a row (I win) or a tail followed by a head (you win). What is the probability that you win?

Chances Are

Stage: 4 Challenge Level: Challenge Level:1

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 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.

Possible support

Allow some calculator use so the focus is on the probability calculations without the comparisons of fractions getting in the way.