In old days, a king had a secret problem to solve. To get divine help, he bought
ID: 3150439 • Letter: I
Question
In old days, a king had a secret problem to solve. To get divine help, he bought three geniee lamps:Golden lamp, Silver lamp, and Brass lamp and luckily all the three genies appear. The probabilites that the king's secret problem can be solved by golden, silver and brass genies are: sin(cot^-1 3/4), cos(csc^-1 5/3) and cos(tan^-1 3), respectively. The king kept the three genies independetly away from one another with a fear that they should not fight with each other.
(a) What is the probability that the king's secret problem will be solved by exactly one genie?
(b) Given that the king's secret problem is solved by one genie, but we do not know which genie, what is the conditional probability that it is solved by the golden genie?
Explanation / Answer
First let us simplify the prob parts
P(golden lampe solves) = sin (arccot 3/4) = 4/5 =0.80
(since sin t = 1/sq rt (1+cot^2 t)
P(silver lampe solves) = cos(csc^-1 5/3) = cos (sin inv 3/5) = 4/5=0.80
P(Brass lampe solves) = cos(tan^-1 3) = cos sec inv rt 10/3 = 3/rt 10= 0.9486
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a) Prob that exactly one solves = P(one solves and other two do not solve)
= 0.8(1-0.8)(1-0.9486)+(1-0.8)0.8(1-0.9486)+(1-0.8)(1-0.8)(0.9486)
=0.054392
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b) Prob (solved by golden genie/solved) = 0.8/Prob solved by any one
P(Problem solved) = P(Gold U silver U brass)
= 0.8+0.8+0.9486-(0.64)-(0.7589)-0.7589 +0.6071 (By additional law of prob and also since events are independent)
= 0.9979
Hence conditional probability that it is solved by the golden genie = 0.8/0.9979
=0.8017
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