Question

Three students A, B, and C are enrolled in the same class. Suppose that A attends class 30 percent of the time, B attends class 50 percent of the time, and C attends class 80 percent of the time. If these students attend class independently of each other, what is (a) the probability that at least one of them will be in class on a particular day and (b) the probability that exactly one of them will be in class on a particular day?

Answer 1

(a) 0.93

(b) 0.38

Explanation:

A attends class 30% of the time, so DO NOT attend 70%

B attends class 50% of the time, so DO NOT attend 50%

C attends class 80% of the time, so DO NOT attend 20%

Writing as probabilities:

P(A) = 0.30 and P(A') = 0.70

P(B) = 0.50 and P(B') = 0.50

P(C) = 0.80 and P(C') = 0.20

(a) the probability that at least one of them will be in class on a particular day

Let's call Q the event of none of them be in class on a particular day

Probability of at least one be in class is the complement of none of them be there, so: 1 - P(Q)

P(Q) = 0.7*0.5*0.2 = 0.07

1 - P(Q) = 1 - 0.07 = 0.93

(b) the probability that exactly one of them will be in class on a particular day?

One of them exactly be in class is

A is B not C not or A not B is C not or A not B not and C is, so

P(A).P(B').P(C') + P(A').P(B).P(C') + P(A').P(B').P(C)

0.3*0.5*0.2 + 0.7*0.5*0.2 + 0.7*0.5*0.8 =

0.03 + 0.07 + 0.28 = 0.38