Question

In Exercises 1–10, use the data in the accompanying table and express all results in decimal form. (The data are from “Mortality Reduction with Air Bag and Seat Belt Use in Head-On Passenger Car Collisions,” by Crandall, Olson, and Sklar, American Journal of Epidemiology, Vol. 153, No. 3.) Drivers Involved in Head-On Collision of Passenger Cars.

Drivers Involved in Head-On Collision of Passenger Cars

	Driver Killed	Driver Not killed
Seatbelt Used	3655	7005
Seatbelt not Used	4402	3040

Complement If A represents the event of randomly selecting one driver included in the table and getting someone who was not killed, what does $\overline{A}$ represent? Find the value of $P\left(\overline{A}\right)$.

Short answer

$\overline{A}$ represents the event of randomly selecting the driver who was killed in the head-on collision.

The value of $P\left(\overline{A}\right)$is 0.445.

Step-by-step solution

Given information

The different categories in the table state the counts of drivers who met a head-on collision.

Define a complement event and its probability

A complementary event of an event E is represented by the symbol that expresses the idea that event E does not occur in the experiment.

The probability ofthe complementary event is stated below.

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

State the complementary event and probability

Add the sum total for the rows and columns.

	Driver Killed	Driver Not killed	Total
Seatbelt Used	3655	7005	10660
Seatbelt not Used	4402	3040	7442
Total	8057	10045	18102

Define event A that a randomly selected driver was not killed in the collision.

As a result, $\overline{A}$ is the event that the randomly selected driver was killed in the passenger car.

Compute the probability 

The number of drivers who were not killed is 10045.

The total number of drivers is 18102.

The probability that the randomly selected driver was not killed is:

$$\begin{gathered} P\left(A\right)=\frac{10045}{18102} \\ =0.555 \end{gathered}$$

The probability that the randomly selected driver was killed is:

$$\begin{gathered} P\left(\overline{A}\right)=1-P\left(A\right) \\ =1-0.555 \\ =0.445 \end{gathered}$$

Thus, the complementary event has a probability of 0.445.