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## Homework Statement

Lars is vacationing in Las Vegas and goes into the largest casino he comes across.

He sits down at the blackjack table and decides to play until he wins and to double

the bet for each game. He bets one dollar in the first game, two dollars in the

second game, and so on until he wins.

Assume (for simplicity) that he always gets back twice what he bet if he wins, that

his probability of winning is 0.3 in every game, and that Lars stops playing after

winning once.

a) Let X be the number of times Lars player before he quits. What is the

probability distribution of X?

b) Assume in the rest of this problem that Lars runs out of money and thus

stops playing if he has not won after playing five games. Let W be the

number of dollars Lars wins at the casino (regardless of how much he has

bet). What is the expected value of W?

c) Let Y be the winnings Lars is left with after his bets are deducted. What is

the expected value of Y ?

## The Attempt at a Solution

I've done a) ##X##~## f(x) = 0.7^{x-1} 0.3 ##, and b) ##W = 2^{X-1} \cdot 2 \Rightarrow E(W) = \Sigma_{x=1}^{5} f(x) 2^x = 6.59$##.

I'm completely stuck on c though. I keep getting the wrong answer, as I try to calculate the expected loss. For instance, why is the following wrong?

##E(L) =##

##+ 0.3\cdot (0)##

##+ 0.7 \cdot 0.3 \cdot (1)##

##+ 0.7 \cdot 0.7 \cdot 0.3 \cdot (1+2)##

##+ 0.7 \cdot 0.7 \cdot 0.7 \cdot 0.3 \cdot (1+2+4)##

##+ 0.7 \cdot 0.7 \cdot 0.7 \cdot 0.7 \cdot 0.3 (1+2+4+8)##

##+ 0.7\cdot 0.7 \cdot 0.7 \cdot 0.7 \cdot 0.7 \cdot (1+2+4+8+16)##

##= 7.5$##

I mean, these are all the possibilities if Lars is only playing a maximum of 5 times?

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