Succession in Randomia

Why do this problem?
Here you have to interpret the information given in the problem, decide whether or not the method of choosing the new king seems fair, and decide what methods you might use to check on whether it is fair or not. It is a novel context and yet easy to understand what the problem is. No resources are needed to run trials except some coins.

Then you need to decide what the possibilities are, what the probability space is and what would constitute a 'trial' to test the relative frequencies of the different possible outcomes.

The problem can be tackled at different levels. It can be treated as an investigation into the experimental probability. Alternatively, learners can draw a tree diagram and calculate the probabilities using a computer program or spreadsheet. Those who have met geometric series should be able to calculate the theoretical probabilities.

Possible approach
Why not start with a class referendum on whether the King's method is fair? Let the learners discuss it in pairs and then as a class and then take a vote. Many learners will think it is not fair because they think Lotto has a better chance as he wins the throne if he gets either two heads or two tails whereas his brothers only win with one or the other.

You could make each pair of learners run 10 or more trials, one tossing the coin until two successive heads or two successive tails come up, and the other recording the results and which son would be king for each trial. Then you could put all the class results together and calculate the relative frequencies. Is the relative frequency for selecting each son approximately one third?

Commonly learners find this problem hard because the probability space has infinitely many events. The class can discuss what the events are and how they fall into four sets of events.

The next task could be to draw a tree diagram and to use this to help to calculate the probabilities after 2, 3, 4, 5, 6, 7, 8 ... events so as to identify the pattern emerging.

A spreadsheet helps with these calculations. Considering only up to 8 tosses the probabilities are (to 4 decimal places): Bingo 0.3320, Toto 0.3320, Lotto 0.3281, still undecided 0.0078. Notice that the probabilities get closer and closer to one third and after 8 tosses you can go on to find out how that 0.0078 is distributed between the three brothers.

For learners who can sum an infinite geometric series, the calculation of the theoretical probability is easy as it only involves powers of one half.

Key questions
What are the possible outcomes?
Could the coin tossing go on for ever without a decision being reached? How likely is that? Why?
What are the probabilities for each son being chosen after two coin tosses?
If no decision is reached after two tosses what are each son's chances with 3 tosses?
If no decision is reached after three tosses what are each son's chances after 4 tosses?
Can you draw a tree diagram?
Can you use a spreadsheet to help with the calculations?
Can you write down the series that you would need to sum to find the probability of Bingo being chosen?
What are the total theoretical probabilities of each of the 4 possible outcomes: Bingo being chosen, Toto being chosen, Lotto being chosen, the coin tossing going on for ever?

Possible extension
For many learners this problem will only involve testing the relative frequencies. For others the extension will be calculating the theoretical probabilities.The problem Rain or Shine requires similar reasoning and is a good one to follow Succession in Randomia.

Possible support
For those who are convinced that Lotto has a better chance and that the relative frequencies should not turn out all to be approximately one third for a large number of trials, suggest that they consider the answers to the key question: "Bingo and Toto are always ahead when there have been an even number of tosses and Lotto's chances only 'catch up' when the next toss is made to give an odd number of tosses, how then can Lotto ever be ahead?"