14. A carnival game consists of hitting a lever with a sledge hammer to propel a

Question

14. A carnival game consists of hitting a lever with a sledge hammer to propel a weight upward toward a bell. Because the hammer is quite heavy, the chance of ringing the bell declines with the number of attempts; in particular, the probability of ringing the bell on the ith attempt is (3/4). For a fee, the carnival sells you the privilege of swinging the hammer until the bell rings or until you have made three attempts, whichever occurs first. (a) Find the probability distribution of X, the number of hits taken. (b) The prize for ringing the bell on the ith try is $(4-1), i = 1, 2, 3, How much should the carnival charge for playing the game if it wants an expected profit of $1 per customer?

Explanation / Answer

14. A carnival game consists of hitting a lever with a sledge hammer to propel a weight upward toward a bell. Because the hammer is quite heavy, the chance of ringing the bell declines with the number of attempts; in particular, the probability of ringing the bell on the ith attempt is (3/4). For a fee, the carnival sells you the privilege of swinging the hammer until the bell rings or until you have made three attempts, whichever occurs first. (a) Find the probability distribution of X, the number of hits taken. (b) The prize for ringing the bell on the ith try is $(4-1), i = 1, 2, 3, How much should the carnival charge for playing the game if it wants an expected profit of $1 per customer?

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