CJC Tutorial Q8

A bag contains \(4\) red, \(5\) blue and \(6\) green balls. The balls are indistinguishable except for their colour. A trial consists of drawing a ball at random from the bag, noting its colour and replacing it in the bag. A game is played by performing \(10\) trials in all.

The player receives \(\$0.50\) for every blue ball drawn in a game. He receives nothing if balls of other colours are drawn.

1. Find the probability that, in a game, he receives
   1. exactly \(\$3\),
   2. more than \(\$2.50\),
2. Find also the amount a player is most likely to receive.
3. At the start of a tournament, each player plays the above game once. Players who earned more than \(\$k\) proceed to the next round. Find the least value of \(k\) such that, in a random sample of \(10\) players, the probability that all the \(10\) players proceed to the next round is less than \(0.1\).