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## 'Succession in Randomia' printed from http://nrich.maths.org/

King At the first of Randomia has a problem. He is the proud father
of three sons: Bingo, Toto and Lotto. Only one of the sons can
succeed him as king but they are all suitable to rule the kingdom.
At his wit's end he decides to use the following method to decide
who his successor will be:

At a succession ceremony, the king will toss a coin repeatedly
until two consecutive heads or two consecutive tails come up.

Bingo will become king if it is two successive heads (...HH) and
this occurs in an even number of tosses.

Toto will become king if it is two tails (...TT) and this occurs in
an even number of tosses.

Lotto will become king if it is either two heads (...HH) or two
tails (...TT) and this occurs in an odd number of
tosses.

The subjects are in an uproar because they think this method of
selection is not fair.

Investigate the situation and decide whether or not the sons have
an equal chance of becoming the next king. Talk to other people
about it, people tend to disagree about this one so you need to
have good reasons for what you think. How could you test out your
theories?

(Thanks to Bertus van Etten and
the Institute for Mathematics and Science Teaching of the
University of Stellenbosch for permission to publish this
problem.)