### Rain or Shine

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

### Knock-out

Before a knockout tournament with 2^n players I pick two players. What is the probability that they have to play against each other at some point in the tournament?

### Squash

If the score is 8-8 do I have more chance of winning if the winner is the first to reach 9 points or the first to reach 10 points?

# Succession in Randomia

##### Stage: 5 Challenge Level:

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.)