discrete math - binomial coefficients

Question

a restaurant needs at least one cook and at least one waitress to open. The owner hires three cooks and three waitresses, each of whom is equally likely to show up for work or call in sick. What is the probability that the restaurant will operate on any given day?

the answer is 49/64 I am just not sure how to get here. Thanks!

Answer 1

Let C be the event of a least one cook showing up, and let W be the event of at least one waitress showing up.

We want to find Prob(C and W). Since these events are independent, we have

Prob(C and W) = Prob(C)·Prob(W).

The easiest way to calculate Prob(C) is to calculate the probability of no cook showing up, Prob(not C), since

Prob(C) = 1 - Prob(not C).

The probability of each cook not showing up is 1/2, so the probability of all 3 cooks not showing up is (1/2)³, since the events are also independent. Therefore

Prob(C) = 1 - (1/2)³ = 1 - 1/8 = 7/8.

Similar calculations give us the probabilities for at least one waitress showing up:

Prob(W) = 1 - Prob(not W) = 1 - (1/2)³ = 1 - 1/8 = 7/8.

And so we conclude that

Prob(C and W) = Prob(C)·Prob(W) = 7/8 · 7/8 = 49/64.