91Ó°ÊÓ

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:

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

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:

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

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.