Laura is feeling lucky and decides to drive down to the Plai

Laura is feeling lucky and decides to drive down to the Plainville casino to play the slot machines. She finds a machine that requires $1 tokens to play, and pays out $40 on every jackpot. Of course, this machine is well-designed by electrical engineers so that the probability of a jackpot is 0.02 on every spin, and the outcomes of the spins are independent. Laura has brought a very large stash of tokens, so she has decided to play until she wins once, and then walk away. Let X be a discrete random variable denoting the number of times she plays the machine. What type of random variable is X How many times will Laura play the machine, on average What is the probability Laura will play more than 4 games Assuming Laura wins on the X-th play. Then, she will have spent $X dollars, and collected $40, so the net winnings are W = 40 - X. Calculate the expected winnings E[W]. Calculate the variance of the expected winnings Var[W].

Solution

a) Her the random variable is the number of trials to get a success. This is follows geometric distribution.

b) Number of times Laura play the maching on an average = 1/p = 1/0.02 = 50

c) Probability function of geometric distribution = p*q^(k-1) where k=1,2,...

Probability laura play more than 4 games = P(x>4) = 1-P(x<=4) = 1-[0.02+0.0196+0.019208+0.018824]=0.922368

d)EW) = 40-E(X) = 40-50= -10

e)Var(W) = Var(40-X) = Var(X) = 0.98/(0.02)^2 (since Var(X) = q/p^2

Var (W) = 2450