4.20. A gambling book recommends the following "winning strategy" for the game of roulette: Bet $1 on red. If red appears (which has18), then take the $1 profit and quit. If red does not appear and you lose this bet (which has probability20of occurring),38probability38make additional $1 bets on red on each of the next two spins of the roulette wheel and then quit. Let X denote your winnings when youquit.a. Find P{X> 0}.b. Are you convinced that the strategy is indeed a "winning" strategy? Explain your answer!c. Find E[X

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Asked Oct 29, 2019
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4.20. A gambling book recommends the following "winning strategy" for the game of roulette: Bet $1 on red. If red appears (which has
18
), then take the $1 profit and quit. If red does not appear and you lose this bet (which has probability
20
of occurring),
38
probability
38
make additional $1 bets on red on each of the next two spins of the roulette wheel and then quit. Let X denote your winnings when you
quit.
a. Find P{X> 0}.
b. Are you convinced that the strategy is indeed a "winning" strategy? Explain your answer!
c. Find E[X
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4.20. A gambling book recommends the following "winning strategy" for the game of roulette: Bet $1 on red. If red appears (which has 18 ), then take the $1 profit and quit. If red does not appear and you lose this bet (which has probability 20 of occurring), 38 probability 38 make additional $1 bets on red on each of the next two spins of the roulette wheel and then quit. Let X denote your winnings when you quit. a. Find P{X> 0}. b. Are you convinced that the strategy is indeed a "winning" strategy? Explain your answer! c. Find E[X

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Expert Answer

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Step 1

X- be the winnings when we quit.
Case-1 win in first spin
Probability of win in first spin = 18/38
profit when you quit X = $1

Case-2 loss in first spin
probability of loss in first spin = 20/38

Possible outcomes in next two spins are (win, win), (win loss), (loss,win),(loss,loss)
probability of win in third spin  and win in fourth spin= (18/38)(18/38)
profit when you quit X = -1+1+1 = 1

probability of win in third spin  and loss in fourth spin= (18/38)(20/38)
profit when you quit = -1+1-1 = -1

probability of loss in third spin  and win in fourth spin= (20/38)(18/38)
profit when you quit = -1-1+1 = -1

probability of loss in third spin  and loss in fourth spin= (20/38)(20/38)
profit when you quit = -1-1-1 = -3

Possible outocmes of X are 1,1,-1,-1,-3 with probabilities as shown below

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х 1 -1 1 -3 (20/38)(18/38)^2 (20/38)^2(18/38) (20/38)^2(18/38) РX) i8/38 (20/38)^3

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Step 2

P(X>0) is calculated as shown below,
we get P(X > 0) = 0.592 (rounded to three decimals).

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18 20 18 P(X> 0)= 38 38 38 P(X>0) 0.473684+0.1180930 = 0.591777

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Step 3

The probability of net profit greater than 0 is 0.592. If we consider this probability we can say that the chance to quit the roulette table with winning will be favourable as it is more than...

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