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At Least One...

Age 11 to 14 Challenge Level:

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

This printable worksheet may be useful: At least one....

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 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.

Possible extension

Same Number! provides a natural extension to this problem.