(10) There is a state fair game called Milk Can that was popular in the last cen

Question

(10) There is a state fair game called Milk Can that was popular in the last century. The player is given three attempts to toss a softball into a 10-gallon metal milk can from a line six feet from the can. Suppose the probablity of a ball getting into the can is 0.4. Let X be the number of times a ball going into the can in a game. (a) Give the probability distribution for X (b) Assume all players have the same capability of getting the ball into the can (40% of (c) Each game costs $1 to play, and the player win $0.90 each time the ball goes into the the average amount won by a player per game if this game is repeated many times. the time.) Find the average number of times the ball going into the can per game. milk can. Let Y be the amount won by a player in a game. Then, Y 0.9X-1. Find (d) Calculations show that the variance for the number of times the ball going into the milk can is 0.48. Find the variance for the amount won by a player in a game.

Explanation / Answer

a) X is binomally distributed random variable, of the form B(n, p), where n = no. Of trials = 3, and p = Probability of success = 0.4

Therefore, X ~ B(3, 0.4)

b) avg no. Of times ball goes in can per game = E(X)

=> E(X) = P(X = 0)*0 + P(X = 1)*1 + P(X = 2)*2 + P(X = 3)*3

= 0 + 1.(3C1)*0.4*(0.6)^2 + 2*(3C2)*(0.4^2)*(0.6) + 3*. (3C3)*(0.4^3)

= 3*0.4*0.36 + 2*3*0.16*0.6 + 3*0.064

= 1.2

Or, we can directly write, for a binomally distributed random variable, E(X) = np = 3*0.4 = 1.2

c) avg ant won = E(Y) = E(0.9X - 1)

= E(0.9X) - E(1)

= 0.9*E(X) - 1

= 0.9*1.2 - 1

= 1.08 - 1

= 0.08

d) Var(Y) = Var(0.9X - 1)

= (0.9^2)Var(X) - Var(1) ........ ( Var(aX + b) = (a^2)*Var(X))

= 0.81*0.48 - 0

= 0.3888

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