1. Bruce is a gambler and regularly plays several rounds of a gamble in which he
ID: 3205369 • Letter: 1
Question
1. Bruce is a gambler and regularly plays several rounds of a gamble in which he wins $15,000 if the head comes on top twice when a fair coin is flipped twice and loses $5,000 for any other outcome from the two flips of the coin. In other words, the outcome of the gamble is determined by flipping a coin twice on each round of the gamble and Bruce wins only if head comes on top on both flips of the coin and loses if any other outcome occurs. Bruce is considering to play 12 rounds of such a gamble next week hoping that he will win back the $15,000 he lost in a similar gamble last week. X represent the number of wins for Bruce out of the next 12 rounds of the gamble.
a. Please calculate the probability of success for Bruce on each round of the gamble. Show how you arrived at your answer.
b. What is the probability that Bruce will win none of the 12 rounds of the gamble?
c. What is the probability that Bruce will lose less than 6 of the 12 rounds of the gamble? Show your work.
d. What is the probability that Bruce will win at least 4 out of the 12 rounds of the gamble? Show your work.
e. What is the probability that Bruce will lose more than 8 of the 12 rounds of the gamble? Show your work.
f. What is the probability that Bruce will lose no more than 3 out of the 12 rounds of the gamble? Show your work.
g. How much money is Bruce expected to win in the 12 rounds of the gamble? How much money is he expected to lose? Given your results, do you think Bruce is playing a smart gamble? Please show how you arrived at your results and explain your final answer.
h. Calculate and interpret the standard deviation for the number of wins for Bruce in the next 15 rounds of the gamble. Show your work.
Explanation / Answer
probability of both heads i.e. of winning = 0.25
a)
The probability of success for Bruce on each round of the gamble is that head comes both times
probability of head = 0.5
hence probability of two heads = 0.25
Therefore the probability of success for Bruce on each round of the gamble = 0.5
b)
the probability that Bruce will win none of the 12 rounds of the gamble =
i.e lossing every game
probablity of loosing is = 1 - probability of winning = 1 - 0.25 = 0.75
0.75^12 = 0.031676
c)
the probability that Bruce will lose less than 6 =
loose 1 + loose 2 + loose 3 +loose 4 + loose 5 =
0.25^11*0.75 + 0.25^10*0.75^2 + 0.25^9*0.75^3 + 0.25^8*0.75^4 + 0.25^7*0.75^5
=0.00002163648
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