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You and I play a game involving successive throws of a fair coin. Suppose I pick HH and you pick TH. The coin is thrown repeatedly until we see either two heads in a row (I win) or a tail followed by a head (you win). What is the probability that you win?

The Better Choice

Age 14 to 16
Challenge Level

Why do this problem?

This problem offers an opportunity to explore and discuss two types of probability: experimental and theoretical. The simulation generates lots of experimental data quickly, freeing time to focus on predictions, analysis and justifications.
 

Possible approach

This problem follows on nicely from Cosy Corner

Explain and demonstrate both games by running the interactivities a few times so that students get a feel for the two games, but don't have sufficient results to draw conclusions about the probabilities. Invite students to predict in which game they would expect to win more points, if they played both games the same number of times. 

Allow students time in pairs in which to analyse each game, so that they can decide which is likely to offer them the better chance of winning more points, and emphasise that they will need to be in a position to offer supporting evidence for their decision.

While students are working, circulate and observe the methods being used. Bring the class together and choose individuals who used different methods to explain what they did to the class, recording what they did on the board.

Record their conjectures on the board and then run the interactivity for each game a few hundred times.

Then revisit students' conjectures and discuss which ones matched the experimental data. If no groups had a correct conjecture, then get them to refine their methods in groups, if they can. For the coins, students may count each possible number of heads/tails but not count repeats, for example counting H,H,H,T the same as T,H,H,H. You could guide them around this misconception by having them see four different coins (tails on the 5p is different to tails on the 1p).

For the spinners, the simplest method is to treat the three spinners separately, so that on average, they get two points for every six games for each spinner. Alternatively, students might systematically list the possible ways of getting 0, 2, 4 and 6 points. In this case, the spinners game becomes similar to the coins game, especially if you treat the outcomes as {6} and {not 6}.

Bring the students back again for a discussion about what they changed to improve their methods.

Key questions

How many points do you think you would get if you played 16/6/36/216 times?

What counts as a different outcome?
 

Possible support

This problem could be tackled as a follow-up to Cosy Corner

Teachers may want to use this recording tool to gather the results of other similar experiments that their students are carrying out:



 

Possible extension

A follow-up problem could be Odds and Evens Made Fair