Combining probabilities

If everyone in your class picked a number from 1 to 225, do you think any two people would pick the same number?

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?

It is believed that weaker snooker players have a better chance of winning matches over eleven frames (i.e. first to win 6 frames) than they do over fifteen frames. Is this true?

After transferring balls back and forth between two bags the probability of selecting a green ball from bag 2 is 3/5. How many green balls were in bag 2 at the outset?

Here are two games you can play. Which offers the better chance of winning?

How can we find out answers to questions like this if people often lie?

Investigate the molecular masses in this sequence of molecules and deduce which molecule has been analysed in the mass spectrometer.

Can you devise a fair scoring system when dice land edge-up or corner-up?

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?