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 using a seatbelt, 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 did not use a seatbelt.

The value of $P\left(\overline{A}\right)$is 0.411.

Step-by-step solution

Given information

The counts of drivers involved in a head-on collision are categorized.

Define a complement event and its probability

A complementary event denoted as $\overline{E}$ for any arbitrary event E is defined as the event of the non-occurrence of event E.

If E has a probability $P\left(E\right)$, then the complementary event has a probability as follows:

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

Define a complementary event with reference to the study

The totals are computed for each row and column.

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

Define event Aas a randomly selected driver from 18102 uses seatbelt.

The complementary event of A is the non-occurrence of A.

Thus, is the event that the randomly selected driverdoes not use a seatbelt.

Compute the probability 

The number of drivers who use seatbelts is 10660.

The number of drivers in the study is 18102.

Thus, the probability of event A is computed as follows:

$$\begin{gathered} P\left(A\right)=\frac{10660}{18102} \\ =0.589 \end{gathered}$$

The probability of the complementary event is obtained as follows.

$$\begin{gathered} P\left(\overline{A}\right)=1-P\left(A\right) \\ =1-0.589 \\ =0.411 \end{gathered}$$

Thus, the complementary event has a probability of 0.411.