George has put three coins in a line on the table and covered them up.

They may have heads up (H) or tails up (T).

When he uncovers them what is most likely, that he will see three heads or two heads and a tail?

This problem explores probability as students find if one outcome is more or less likely to occur than another outcome.

Probability is an important concept in our society. It is the basis for the insurance and gambling industries as well as underpinning many crucial business and military decisions.

During this lesson students develop ways of being systematic in their presentation and organisation of material. This skill is important in situations where events are counted and we must ensure that they are counted only once. It is a useful general skill in much of mathematics (and in life) and ensures that nothing vital is missed or overlooked.

### Problem

George has put three coins in a line on the table and covered them up. They may have heads up (H) or tails up (T). When he uncovers them what is most likely, that he will see three heads or two heads and a tail?

### Teaching Sequence

- Introduce the subject of coins and heads and tails.

*Why do captains spin coins at the start of a hockey or cricket match?*

What is more likely, to get a head or a tail?

If you tossed a coin ten times what is more likely, that you would get 10 heads or 5 heads and 5 tails? - Provide coins for students to experiment, first emphasizing the importance of keeping track of their results.
- Have students discuss and compare results, noticing variation. Highlight the language of 'more likely', 'less likely'. Comment on recording that is systematic and organized.
- Ask: If I put two coins under this piece of cloth, what numbers of heads and tails might I see?

*How many ways might I get two heads? (1)*

How many ways might I get a head and a tail? (2)

How could I record this so I don’t forget any possibilities? - Read George's problem. Revisit the need to keep track.
- As students work on the problem ask:

*How many ways can George get three heads?*

How can you record that?

How many ways can he get two heads and a tail? - Any group that finishes early can try the Extension problem.
- Have students report back to the class. Emphasize the language of ‘least likely’, ‘most likely’ and ‘equally likely’.

### Extension

What other possibilities are there for George’s coins? Say which are least likely, most likely and which are equally likely.

### Solution

It is more likely that George will have two heads and a tail than three heads: HHT, HTH and THH compared with HHH. Alternatively it is least likely that he will Have three heads.

#### Extension:

The full range of possibilities is shown below.

HHH;

HHT, HTH, THH;

HTT, THT, TTH;

TTT.

Two heads and a tail and two tails and a head are more likely than three heads or three tails. On the other hand, three heads and three tails are equally likely. Similarly, two heads and a tail and two tails and a head are equally likely. Three heads (or three tails) are less likely than two tails and a head (or two heads and a tail).

(Note the systematic way that the various possibilities are written down. Start by considering heads. There is clearly only one arrangement that will give three heads. Next consider how we could have one tail and then try only one head. Finally consider all tails (or no heads).