Suppose you are playing a simple game that costs you $5 per play, where you draw one cardfrom a well-shuffled deck of 52 cards.• If the card is a spade, you receive $20. (You don’t get the $5 back.)• If it is not, then you do not receive any money. (You don’t get the $5 back.)Let X = the amount of money you have before playing a round of this “game”.Let Y = the amount of money you have after playing a round of this “game”.(a)  Compute the conditional expectation of Y given X = $100.(b)  You can also use the following variant of the Law of Total Expectation tocompute a conditional expectation:E(Z | X = x) = Summation(E(Z | Ai ∩ X = x)P (Ai | X = x)For this part, let Z = the amount of money you have after playing TWO rounds ofthis “game”. Use the above formula to calculate the conditional expectation of Z givenX = $100. (Note: the cards are well-shuffled again between rounds.)(c)  Now suppose you get only $16 for drawing a spade. Compute the conditionalexpectation of Y (not Z) given X = $100

A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Suppose you are playing a simple game that costs you $5 per play, where you draw one card
from a well-shuffled deck of 52 cards.
• If the card is a spade, you receive $20. (You don’t get the $5 back.)
• If it is not, then you do not receive any money. (You don’t get the $5 back.)
Let X = the amount of money you have before playing a round of this “game”.
Let Y = the amount of money you have after playing a round of this “game”.
(a)  Compute the conditional expectation of Y given X = $100.
(b)  You can also use the following variant of the Law of Total Expectation to
compute a conditional expectation:
E(Z | X = x) = Summation(E(Z | Ai ∩ X = x)P (Ai | X = x)
For this part, let Z = the amount of money you have after playing TWO rounds of
this “game”. Use the above formula to calculate the conditional expectation of Z given
X = $100. (Note: the cards are well-shuffled again between rounds.)
(c)  Now suppose you get only $16 for drawing a spade. Compute the conditional
expectation of Y (not Z) given X = $100

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