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

Both Using Seatbelts If 2 drivers are randomly selected without replacement, find the probability that they both used seatbelts.

Short answer

The probability that two randomly selected drivers, chosen without replacement, used seatbelts is 0.347.

Step-by-step solution

Given information

The values for counts of drivers are tabulated under different categories.

Define multiplication rule

When two events take place simultaneously, the probability of joint occurrences is equivalent to the product of their individual probabilities.

In case the events are dependent, the probabilities change with each successive occurrence. The condition when selections are made without replacement describes the occurrence of dependent events.

Step 3:Compute the additive total row-wise and column-wise

Sum up all row and column totals.

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

Compute the probability using the multiplication rule

Define the events as follows:

E: The first randomly selected driver use a seatbelt

F: The second randomly selected driver use a seatbelt

A: The two drivers randomly selected without replacement use seatbelts

The number of drivers who use seatbelts is 10660.

The total number of drivers surveyed is 18102.

Thus, the probability that the first driver selected uses a seatbelt is:

$P\left(E\right)=\frac{10660}{18102}$

The number of drivers left who use seatbelts, after selecting the first driver, is 10659.

The total number of drivers left after the first selection is 18101.

The probabilities that the second driver uses a seatbelt is:

$P\left(F\right)=\frac{10659}{18101}$

The probability that two drivers randomly selected without replacement use seatbelts is:

$$\begin{gathered} P\left(E\text{and}F\right)=P\left(E\right)\times P\left(F\right) \\ =\frac{10660}{18102}\times \frac{10659}{18101} \\ =0.347 \end{gathered}$$

Thus, the probability that two drivers randomly selected without replacement use seatbelts is 0.347.