probability of at least one secretary absent, at least one secretary comes to work and only one comes to work

Question

A businessman has two secretaries. The probability that the one hired most recently will be absent on any given day is 0.08 and the probability that the other secretary will be absent on any given day is 0.07. The probability that both secretaries will be absent on any given day is 0.01.

1) What is the probability that at least one of the secretaries is absent on any given day?

2) What is the probability that at least one of the secretaries comes to work on any given day?

3) What is the probability that only one of the secretaries comes to work on any given day?

Answer 1

From the givens it is clear that the secretaries are not independent in their absences. You can make a Venn diagram: one circle is .08, the other .07, and they overlap .01.

1) Probability of at least one absent will be .08 + .07 - .01. = .14 (P(S1) + P(S2) - P(S1∩S2)

2) P(at least one at work) = 1-P(both absent) = .99

3) P(only one at work) = P(at least one absent) - P(both absent) = .14 - .01 = .13