### 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?

# Magic Bag

##### Age 16 to 18Challenge Level

A magic bag contains some black and white balls, all of the same size and shape.

I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly $0.5$.

I wonder to myself how this condition constrains the numbers of black and white balls.

At the same time, the bag, who can read minds, says "I'll give you a hint: it's got something nice to do with triangle numbers."

Did you know ... ?

Probability theory has many overlaps with the study of combinatorics. Is it fascinating how beautiful and tricky the consequences of counting become!