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?

Answer 1

a) Since the probability of each attempt is of the order (3/4)^i

We have

X	1	2	3
P(X)	0.75	0.5625	0.421875

ie P(X=2)=(3/4)^2

P(X=3)=(3/4)^3

The above table is the probability distribution of X as there can be a maximum of 3 tries

b)

Now given that the prize for ringing the bell= 4-i

So first we find the expected prize that the carniva game will need to give

We need to add 1 dollar to the expected prize so that the owners earn a expected profit of 1 dollar

X	1	2	3
Prize	3	2	1
P(X)	0.75	0.5625	0.75
Expected Amount for the prize	4.125

The expected amount is calculated by multiplying the probability P(X) with the Prize amount

Hence the amount the carnival game has to charge is 4.125+1=5.125 dollar to earn 1 dollar expected profit