Suppose that the joint distribution of the random variables X and Y is given by

Question

Suppose that the joint distribution of the random variables X and Y is given by the following table: What is the probability that X equals 5 given that Y equals 1? Consider a manufacturing process of engine valves. All valves are subject to a first grind. If a valve's thickness meets specifications, then it is ready for installation. If the thickness is too large, then the valve is reground. If the thickness is too small, then the valve is scrapped. Assume that after a first grind, 75% of the valves meet the specifications, 15% are reground and 10% are scrapped. Furthermore. assume that among the valves that are reground, 95% meet the specifications, and 5% are scrapped. Find the probability that a valve will meet the specification (after either the first or the second grind). Assume that each of your calls to a popular radio station has a probability of 0.02 of connecting (that is not obtaining a busy signal). Assume that your calls are independent. What is the probability that your first call that connects is your tenth call? 0.0741 0.0107 0.817 0.183 0.1677

Explanation / Answer

1.

P(x=5|y=1) = (7/45)/(8/45+6/45+7/45) = 0.333333333 [ANSWER, D]

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2.

Let

S = eventually meets specifications
S1, S2 = meets specifications on first, second grind
R = reground
X = scrapped

Hence,

P(S) = P(S1) + P(R) P(S2|R) = 0.75 + 0.15*0.95 = 0.8925 [ANSWER, B]

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3.

Here,

p = probability that you will connext = 0.02
q = probability that you will not connect = 1 - 0.02 = 0.98

Hence,

P(10th call) = 0.98*0.98*0.98*0.98*0.98*0.98*0.98*0.98*0.98*0.02 = 0.016674955 = 0.0167 [ANSWER, B]

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