Three Coins on a Table

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

1. Introduce the subject of coins and heads and tails.
   Why do captains spin coins at the start of a hockey (football) 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? (Do the experiment.)
2. If I put two coins under this piece of cloth, what numbers of heads and tails might I see? (Do the experiment.)
   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?
3. Let the students work in their groups to solve George’s problem. Emphasise that the coins are in a row.
4. Check the progress of each group. Give assistance where needed.
   How many ways can George get three heads?
   How can you record that?
   How many ways can he get two heads and a tail?
5. Any group that finishes early can try the Extension problem.
6. Get the students to report back to the whole class. Make sure that they understand ‘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

One way to do this problem is to draw all the possible outcomes. In this case for three heads we have: HHH. For two heads and a tail we have HHT, HTH and THH. So it is more likely that George will get two heads and a tail than three heads. Alternatively it is least likely that she will have three heads.

Extension:

The full range of possibilities is shown below.

HHH;
HHT, HTH, THH;
HTT, THT, TTH;
TTT.

From that we see that 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).