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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:

A wonderful solution to this problem was sent in by Mark from British School of Manilla, which we have included as a pdf for easy viewing

See Mark's Solution.

This solution is included as submitted and needed no editing or correcting. Well done Mark!
Steve says this about his problem:

There are plenty of worse X-die, the worst being (1, 1, 1, 1, 1, 16). (It will be 'obvious' but a little awkward to demonstrate this. Comparison with (1, 1, 1, 1, 2, 15) is a start) .

It transpires that if you use only the numbers 1 to 6 then no single X-die is better or worse than an ordinary die, with $P(A> B)=P(B> A) = \frac{5}{12}$.

However, one X-die with the numbers 1 to 6 can be better than another.

For example, A = (2, 2, 2, 3, 6, 6) and B = (1, 4, 4, 4, 4, 4).

For these we have $P(A> B) = \frac{4}{9}$ and $P(B> A) = \frac{5}{9}$.