Question

Sobriety Checkpoint When the author observed a sobriety checkpoint conducted by the Dutchess County Sheriff Department, he saw that 676 drivers were screened and 6 were arrested for driving while intoxicated. Based on those results, we can estimate that $P\left(I\right)$= 0.00888, where I denotes the event of screening a driver and getting someone who is intoxicated. What does$P\left(\overline{I}\right)$ denote, and what is its value?

Short answer

$P\left(\overline{I}\right)$represents screening a driver who is not intoxicated. Its value is equal to 0.99112.

Step-by-step solution

Given information

The probability of screening a driver who is intoxicated is denoted by $P\left(I\right)$and is equal to 0.00888.

Define complementary events

Let E be an event. Then, represents the event “not E.”

$\overline{E}$is called thecomplementof E and vice-versa.

The below property is followed by two complementary events:

$P\left(E\right)+P\left(\overline{E}\right)=1$

Define and compute the probability of a complementary event

Let I be the event of screening a driver who is intoxicated.

Therefore, $\overline{I}$is the event of all outcomes that are not included in event I.

Thus, the probability of the complement of the event I is expressed as follows:

$$\begin{gathered} P\left(\overline{I}\right)=1-P\left(I\right) \\ =1-0.00888 \\ =0.991 \end{gathered}$$

Thus,$P\left(\overline{N}\right)$ is equal to 0.991.