At Least One. In Exercises 5–12, find the probability.

At Least One Defective iPhone It has been reported that 20% of iPhones manufactured by Foxconn for a product launch did not meet Apple’s quality standards. An engineer needs at least one defective iPhone so she can try to identify the problem(s). If she randomly selects 15 iPhones from a very large batch, what is the probability that she will get at least 1 that is defective? Is that probability high enough so that she can be reasonably sure of getting a defect for her work?

The probability of selecting a defective iPhone is 0.20.

Fifteen iPhones are selected at random.

The engineer needs at least one defective iPhone.

The probability of an event happening“at least once” is the probability of the eventhappening once or more than once.

It is also equal to one minus the probability of the event not happening at all.

$P\left(\text{A}\text{happening}\text{at}\text{least}\text{once}\right)=1-P\left(\text{not}\text{A}\right)$

Let A be the event of selecting a defective iPhone and be the event of selecting any good iPhone.

The probability of selecting a defective iPhone is given by:

$$\begin{gathered} P\left(A\right)=\frac{20}{100} \\ =0.2 \end{gathered}$$

Thus,P(A) is equal to 0.2.

The probability of selecting a good iPhone is given by:

$$\begin{gathered} P\left(\overline{A}\right)=1-P\left(A\right) \\ =1-0.2 \\ =0.8 \end{gathered}$$

Thus,$P\left(\overline{A}\right)$ is 0.8.

Fifteen iPhones are selected.

The probability of getting all good iPhones or not getting a defective iPhone is given by:

$$\begin{gathered} P\left(\text{all}\text{good}\right)=\left(0.8\right)\times \left(0.8\right)\times ...\times \left(0.8\right) \\ ={\left(0.8\right)}^{15} \\ =0.0352 \end{gathered}$$

The probability of getting at least one defective iPhone is one minus the probability of getting all good iPhones:

$$\begin{gathered} P\left(\text{at}\text{least}\text{1}\text{defective}\right)=1-P\left(\text{all}\text{good}\right) \\ =1-0.0352 \\ =0.965 \end{gathered}$$

Therefore, the probability of getting at least one defective iPhone is equal to 0.965.

The probability of getting at least one defective iPhone is very high; 0.965.

Also, the engineer needs at least one defective iPhone to identify the problem(s).

The probability value of selecting at least one defective iPhone is fairly high. So, the engineer can be sure of getting a defect to work upon.