1. Julia plays the following game: There is a bag full of 10 chips. There are 3

Question

1. Julia plays the following game: There is a bag full of 10 chips. There are 3 win- ning chips and 7 losing chips. Julia is allowed to draw 2 chips (without replacement) For each winning chip she pulls out, she wins $2. However, the game costs $1.50 to play. (So her actual winnings if she pulls out a single winning chip is only $0.50.) Let X be her winnings. (a) Find the expected value and variance for the random variable X (b) Based on your answer to part (a), should she play the game? (c) What is a fair price for this game? That is, how much should Julia be charged for this game for the expected value to be 0?

Explanation / Answer

a) here P(X=-1.5) =P( both are losing chips) =(7/10)*(6/9)=7/15

P(X=0.5) =P(one is losing and other is winning chips) =2*(7/10)*(3/9)=7/15

P(X=2.5)=P( both are winning chips) =(3/10)*(2/9)= 1/15

therfore

from above mean =-0.30

and variance =1.4933

b) as expected value is less then 0 which mean that she is expected to lose 30 cents on average for every bet; she should not play

c) fair value is where expected value is 0.

hene fair price =1.5-0.30 =1.2

x p(x) xP(x) x2P(X) (x-)2 (x-)2P(x) -1.5 7/15 -0.700 1.050 1.440 0.672 0.5 7/15 0.233 0.117 0.640 0.299 2.5 1/15 0.167 0.417 7.840 0.523 total 1 = -0.30 1.583 9.920 2= 1.4933

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