1. You are playing a simple bar game where you roll a ball from one end of the pool table; the ball must bounce against the far side at least once, and the goal is for the ball to stop as close as possible to near side. The pool table is 92 inches long. One player has just had her ball stop 17 inches (from the near side). To win, you must beat that -- i.e. your ball must stop closer than 17 inches from the near side. Little Miss DrunkyPants, filled with exuberance, courage, and utter lack of skill, decides to play. With a shrill "Woo-Hoo!" she rolls the ball with abandon. What is the probability she will win? (In all these cases, you may assume the ball does bounce off the far side at least once, and we allow it to bounce off either side any number of times.)
2. Where would the threshold need to be moved (from its current location at 17 inches) to give Little Miss DrunkyPants a 51% chance of winning?
3. You decide to give Little Miss Drunky Pants 2 tries: she wins if she can beat 17 inches on one or both tries. (You regard these tries as independent, because her current state seems to preclude any improvement of her motor skills, at least until tomorrow morning.) What is the probability of her winning?
4. You decide to give Little Miss DrunkyPants 5 tries (and you again assume independence among the tries). What are the chances that she will beat 17 inches on 3 of those 5 tries?
5. Given your assumptions about Little Miss DrunkyPants, draw the pdf corresponding to the probability distribution associated with her score (in inches from the near side of the pool table).
6. Sally Swillwell is highly skilled at this game (regardless of the threshold, which in this example, happens to be 17 inches). Draw a pdf curve that might correspond to the distribution of her score. Make your curve so that there is a unique mode (i.e. an x such that f(x) is at its largest). Be sure to label the x-axis. [Hint: There are many correct (and incorrect) ways to do this; the key is to draw a continuous pdf that reflects Sally's skill].