Question

A warehouse contains ten printing machines, five of which are defective. a company selects four of the machines at random, thinking all are in working condition. what is the probability that all four machines are nondefective? (round your answer to four decimal places.)

Answer 1

This kind of situation is modelled by Bernoulli's formula. It applies everytime there is an experiments with two possible outcomes repeated n times. Each repetition is independent on the others, and we know the probability of the two outcomes p and 1-p. If we want the outcome with probability p to appear k times, the probability is

$\binom{n}{k}p^k(1-p){n-k}$

In your case, you run the "experiment" 4 times (you choose 4 printers) and want that all of them to be non-defective. A printer is non-defective with probability 1/2, since there are 5 defective and 5 non-defective printers.

So, our model is built with n = k = 4, p = 1-p = 1/2. The probability is

$P=\binom{4}{4}\left(\cfrac{1}{2}\right)^4\left(\cfrac{1}{2}\right)^0 = 1 \cdot \left(\cfrac{1}{2}\right)^4 \cdot 1 = \cfrac{1}{16}$