Are coins fair?

Materials

· Coins

· Calculators

· Spreadsheet (optional)

· Scripts for the opening scene of Rosencrantz and Guildenstern are Dead by Tom Stoppard (optional)

Activities

1. Have two students read the opening scene from R&G are dead (if the class has a suitable reading level). Alternately, stand at the front of the room and flip a coin, tallying the results. Claim to always get “heads” until students begin to protest and claim something isn’t right.

2. Discuss with the class how they feel about the scene they have witnessed. Is the run of “heads” possible? Try to encourage a distinction between events that are impossible, and those that are highly unlikely.

3. Create a table (on the chalkboard or spreadsheet):

Number of flips	Probability of all heads
1
2

Ask how likely it is to get heads in one flip of a coin. You may need to introduce the idea of theoretical probability, which is determined by dividing the number of successful events by the total number of possible events. In this case there is one way to get heads, and two possible outcomes, so the theoretical probability is 0.5.

4. Hand out at least eight coins to each student, pair or group. Ask them to place the coins to show all the possible outcomes of flipping two coins. It is likely that some will claim there are three possibilities. With any luck some will claim there are four possibilities. Allow debate. If the “three” side seems to be winning, or if there is no “four” side to begin with, display pairs of two different coins (e.g., a penny and a nickel). Start with two heads, then two tails, then ask what the third possibility is. Ask whether the penny is heads or tails. This might disequilibriate things a bit.

5. In any case, the next step is to suggest determining the probability experimentally. Have each student flip a pair of coins and record the result. Tabulate the results on the board or spreadsheet. If you use a spreadsheet a circle graph representation of the data might be useful in the discussion to follow.

6. Discuss whether the results seems to be equally likely based on their experiment. Ask for suggestions. One might be that the number of flips was too small. (If so, someone understands about sampling; encourage them to explain to the class). Have everyone do another flip (or two) and record the result. Add those results to your tabulation. You might mention that you are tabulating the frequency of each outcome. Reopen the discussion of the results.

7. By now there should be some agreement that the heads+tails result is more likely than heads+heads and tails+tails. Point out that the experimental results don’t come out to the same as the theoretical probabilities, but that with the larger number of coins, it came out closer to the theoretical result.

8. Have the students, in groups, determine all the possible outcomes for three coins. You might suggest they use different colours or something to keep track of which coin is which. Add to the table. Do the same for four coins.

9. Ask if they see any patterns in the table so far that would suggest a prediction for five coins. Either check the prediction or generate more detail by having them work out the possible outcomes for five coins.

10. If they have now predicted that the probability of getting n heads in a row is , have them calculate the probability of the run of heads in R&G or in your opening. They’ll need calculators.

11. Now for the question: Are coins fair? Ask what they would predict would be the result of flipping ten coins. All heads and all tails are unlikely, but what would happen? They may predict that there will be five heads and five tails. Do the experiment (in groups).

12. Tabulate the results. In all likelihood few if any groups will have five of each. Have them try to explain this. If they conclude that more coins would be more likely to be balanced, repeat with 20 coins.

13. For this next bit, it would be nice to have 32 kids. Assign each person one of the 32 possible outcomes for five coins (determined earlier). Suggest that the outcomes they have been assigned are the results of the first five flips of ten coins and they are going to work out what the possible outcomes are for the next five coins. You may not have to do this all the way to the end, if they protest that the second five will be the same as the first. However it works out, eventually raise the question of what the total number of outcomes is (32x32=1024). Then have them count how many of those have the same number of heads as tails (252). So the theoretical probability of equal numbers of heads and tails is 252/1024 (about 1/4).

Outcomes

Grade 4

F2- describe data maxima, minima, range and frequency

G1- predict probabilities as either close to 0, near 1, or near ½

G2- cite examples of everyday events with very high or very low probabilities

G3- predict whether one simple outcome is more or less likely than another

G4- use fractions to describe experimental probabilities

Grade 5

G1- conduct simple experiments to determine probabilities

G2- Determine simple theoretical probabilities and use fractions to describe them

Grade 6

F1- choose and evaluate appropriate samples for data collection

F5- use circle graphs to represent data proportionally

F7- make inferences from data displays

G1- conduct simple simulations to determine probabilities

G2- evaluate the reliability of sampling results

G3- analyze simple probabilistic claims

G4- determine theoretical probabilities

G5- identify events that could be associated with a particular theoretical probability