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Nines and Tens

Explain why it is that when you throw two dice you are more likely to get a score of 9 than of 10. What about the case of 3 dice? Is a score of 9 more likely then a score of 10 with 3 dice?

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Card Game (a Simple Version of Clock Patience)

Four cards are shuffled and placed into two piles of two. Starting with the first pile of cards - turn a card over... You win if all your cards end up in the trays before you run out of cards in one pile. What is the probability that you win?

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Master Minding

Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?

At Least One...

Stage: 3 and 4 Challenge Level: Challenge Level:1

Why do this problem?

An important strategy in answering probability questions requires us to consider whether it is easier to work out the probability of an event occurring or the probability of it NOT occurring. In this problem, learners are introduced to tree diagrams and the concept of mutually exclusive events whose probabilities sum to 1.

Possible approach

Ask the introductory question:
"Imagine flipping a coin three times. What's the probability you will get a head on at least one of the flips?" 
Give the class time to explore on their own or in pairs, then share the different methods they used to work it out.
 
If no-one has suggested a tree diagram, start building a tree diagram on the board and ask for suggestions of how to complete it. Then ask the class to identify which branches contain at least one head, and to use this to work out the probability of getting at least one head.
 
With the class working in groups of three or four, challenge them to build on what they've done by asking them to work out the probability of getting at least one head in four flips. Wander around the class and ask groups to move on to five flips, six flips... as soon as they've finished the one they are working on. As each group discovers a neat way of working out these probabilities, first challenge them to work out the probability of getting at least one head in twenty flips, and then, assuming they can apply their method successfully, give them one of the related questions from the problem.
 
Once most of the groups have a successful method for the at least one head problems, bring the class together to discuss what they noticed when working on their tree diagrams, and to justify the methods they used to work out the probabilities.

Finally, the remaining questions from the problem can be used with the class to consolidate these ideas.

Key questions

What is the probability of getting at least one head?
What is the probability of getting no heads?
How are the probabilities related?

Possible extension

Same Number! provides a natural extension to this problem.
 

Possible support

Spend some time working together as a class on listing probabilities, and then move to the tree diagram representation simply as an efficient way of listing systematically.

 You can read about some of the issues which might arise when teaching probability in this article.