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
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.
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
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?
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.
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?"