DGC value is 458. Answers are provided please show work. 453 Question 2.4. (QUEE
ID: 3361824 • Letter: D
Question
DGC value is 458. Answers are provided please show work. 453 Question 2.4. (QUEEN'S SECRET AND GENIES) In old days, there was a queen and she had a secret problem to solve. To get an answer to her secret problem, the queen bought one female and one male genie lamps. The probabilities that her secret problem can be solved by the male and female genies are 0.6 and 0.74 100 (2DGC+500) respectively. Fig. A Queen looking for divine heip C a) Assume the queen keeps the genies independently away from each other with a fear that both could not harm her, what is the probability that the queen's secret problem is solved by at least one of the genies? o. 96 32 (b) Assume the queen keeps the genies independently away from each other with a fear that both could not harm her, what is the probability that the queen's secret problem is solved by exactly one of the genies? (c) Assume the queen's problem is solved by exactly one genie, but we do not know which one, what is the conditional probability it is solved by the female genie? (d) Assume the queen's problem is solved by at least one genie, what is the conditional probability it is solved by both genies? (e) Assume the queen allows both genies to work together and the probability that at least one solved her problem is 0.95. What is the probability that her genies together? will be solved together by the both PlaUB)0.95Explanation / Answer
P(M) = 0.6
P(F) = 0.7 + 100/ (2*458 + 500 ) = 0.7706
a) Probability that the problem is solved by at least one of them is computed as:
= 1 - Probability that it is solved by neither of them
= 1 - (1 - 0.6)*( 1 - 0.7706)
= 0.9082
b) Probability that it is solved by exactly one of them
= P(M)(1 - P(F) ) + P(F) ( 1 - P(M) )
= 0.6*( 1- 0.7706) + 0.7706*(1 - 0.6)
= 0.4459
c) Given that the problem is solved by exactly one of them, probability that it is solved by female is computed as:
= P(F) ( 1 - P(M) ) / 0.4459
= 0.7706*(1 - 0.6) / 0.4459
= 0.6913
d) Given that it is solved by at least one of them, probability that it is solved by both of them is computed as:
= P(M)P(F) / 0.9082
= 0.6*0.7706 / 0.9082
= 0.5091
e) Here, we are given that:
P(M or F) = 0.95
Using law of total probability, we get:
P(M) + P(F) - P(M and F) = P(M or F)
0.6 + 0.7706 - P(M and F) = 0.95
P(M and F) = - 0.95 + 0.6 + 0.7706 = 0.4206
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