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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.


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?


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


Age 16 to 18
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.