### 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.

### 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?

### Snooker

A player has probability 0.4 of winning a single game. What is his probability of winning a 'best of 15 games' tournament?

# X-dice

##### Stage: 5 Challenge Level:

This problem involves conditional probability.

Consider two dice $A$ and $B$, where the largest number on $A$ is $N$. Then $P(A< B)$ is
$$P(A< B) = \sum^N_{m=1}P(A< B|A=m)P(A=m)$$
Of course, in this expression $P(A=m)$ is zero if the integer $m$ is not present on the die.

With problems such as these, don't be afraid to start with a period of experimentation: just choose any numbers to begin with and explore their properties. The structure of the problem will soon start to emerge.