### 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.