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

Phone Survey Subjects for the next presidential election poll are contacted using telephone numbers in which the last four digits are randomly selected (with replacement). Find the probability that for one such phone number, the last four digits include at least one 0

Four digits are randomly selected with replacement.

The event defined byat least one occurrence of a specific outcome is complementary to the event that no such outcome appears.

Mathematically,

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

Let A be the event of selecting zero from digits 0 to 9.

The complement of event A will be selecting any digit other than zero.

The total number of digits is 10.

$$\begin{gathered} P\left(A\right)=\frac{1}{10} \\ =0.1 \end{gathered}$$

Thus, is 0.1.

The probability of the complementary event is computed as follows:

$$\begin{gathered} P\left(\overline{A}\right)=1-P\left(A\right) \\ =1-0.1 \\ =0.9 \end{gathered}$$

Thus, is equal to 0.9.

In a four-digit set of numbers selected at random, the probability that at least one zero occurs is computed as follows.

The probability of selecting a digit other than zero out of the four digits is given by:

$$\begin{gathered} P\left(\text{not}\text{selecting}0\right)=\left(0.9\right)\times \left(0.9\right)\times \left(0.9\right)\times \left(0.9\right) \\ =0.656 \end{gathered}$$

The probability of having at least one zero is equal to one minus the probability of having no zeros:

$$\begin{gathered} P\left(\text{at}\text{least}\text{one}0\right)=1-P\left(\text{not selecting}0\right) \\ =1-0.656 \\ =0.344 \end{gathered}$$

Therefore, the probability of having at least one zero is equal to 0.344.