The probability of a random selected student coming to class is 5/6. There are a

Question

The probability of a random selected student coming to class is 5/6. There are a group of 3 students A, B and C. Suppose C is an exchange student and does not know anyone. Students A and B are housemates. A comes to class 5/6 of the time, like most students, but also makes an effort to wake and bring along his housemate when A goes to class. This means that B will be in class with probability 0.9 if student A goes, but only 0.3 if student A does not.

a)What is the probability that both A and B come to class?

b)What is the probability that all three students attend class?

c)What is the probability that that B comes to class?

d)If B is in class, what is the probability that A does not come?

e)What is the probability that C comes if B does not?

f)What is the probability that at least one student comes to class?

Now let X be # of 3 students who come to class. X = (0,1,2,3):

g)What is the probability that P(X=x) =p(x) for all x = 0,1,2,3? Verify it is valid distribution.

h)Find the expected value, standard deviation of X and variance?

Explanation / Answer

(A) Probability that both A ad B Come to class = P(A come to class and B come to class) = 5/6 * 0.9 = 0.75

(B) Probability that all three students attend class = P(all 3 students attend class) = 5/6 * 0.9 * 5/6 = 0.625

(C) Probability that B comes to class = P(A come to class) * B(Come to class) + P(A doesnt come to class) * P(B come to class) = 5/6 * 0.9 + 1/6 * 0.3 = 0.8

(D) If B is in class, Prbabolity that A does not come = (1/6 * 0.3)/ (5/6 * 0.9 + 1/6 * 0.3) = 0.0625

(E) What is the probability that C comes if B doesnot = It doesn't affect that if B comes then C will come so the proability is still the same = 5/6

(F) Probability that at least one student comes to class = 1 - Pr(nobody came to class)

Pr(nobody came to class) = 1 - P(A') * P(B') P(C') = 1 - (1/6) * (1-0.8) * (1/6) = 179/180

(G) X = 0,1,2,3

so P(X=0) = 1/180

P(X=1) = P(A)P(B')P(C') + P(A')P(B) P(C') + P(A')P(B')P(C) = 5/6 * 0.2 * 1/6 + 1/6 * 0.3 *1/6+ 1/6 * 0.2 * 5/6 = 23/360

P(X=2) = P(A)P(B) P(C') + P(A)P(B')P(C) + P(A') P(B) P(C) = 5/6 * 0.9 * 1/6 + 5/6 * 0.2 * 5/6 + 1/6 * 0.3 * 5/6 = 55/180

P(X=3) = 225/360( as calculated previously)

(H) E(X) = 0 * 1/180 + 1 * 23/360 + 2 * 55/180 + 3 * 225/360 = 2.55

Varaince(X) = 1/180 * (2.55 - 0)2 + 23/360 * (2.55 -1)2 + 55/180 * (2.55 -2)2 + 225/360 * (2.55 -3)2 = 0.4086

Standrd deviation(X) = sqrt [Var(X) ] = sqrt(0.4086) = 0.64

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