Game winnings, a mode and a tournament cutoff

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.

(a) Find the probability that, in a game, he receives

(i) exactly $3,

(ii) more than $2.50,

(b) Find also the amount a player is most likely to receive.

(c) 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.