answer all questions with explanation

Problem 1. (16 pt) At a carnival game customers buy a bucket of balls and for every ball they succeed in tossing through a hoop they get a crummy little prize. To compensate for supposed differences in skill, men are given a bucket of 12 balls, women a bucket of 15 balls, and children a bucket of 18 balls. But in fact there is absolutely no skill involved in the game; every ball goes through the h00p with probability 1/2 independently of the other balls. In a particular week, 1000 men, 2000 wornen, and 4000 children play the game. During this week, what is the probability that there is exactly one person who doesn't win any prizes. (Compute a numerical answer. You may use an appropriate approximation.)