Succession in Randomia

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.)