06: Problem 11 Previous Problem Problem List Next Problem (2 points) Consider a

Question

06: Problem 11 Previous Problem Problem List Next Problem (2 points) Consider a game played by flipping biased coins where the probability of heads is 0.19. You first choose the number of coins you want to flip. You must pay $0.75 for each coin you choose to flip. You flip all the coins at the same time. You win $1000 if one or more coins comes up heads. How many coins should you flip to maximize your expected profit? Answer: (Your answer should be an integer.) What is your maximum expected profit? Answer: $

Explanation / Answer

for above let n be the number of coins that need to be flipped

probabilty that at least one head in n coins =1-P( no head in n coins) =1-(0.81)n

hence expected profit =P =expected gain -cost of coins =1000*(1-(0.81)n)-0.75n

differntiating above with respect to n:

dP/dn =-1000(0.81)n*ln(0.81)-0.75

putting above equal to 0:

-1000(0.81)n*ln(0.81) =0.75

(0.81)n =0.00355

taking again log

nln(0.81)= ln(0.00355)

n =~27

therefore expected coins to be flipped =27

maximum expected profit =$976.37

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