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 Drivers Killed If 2 drivers are randomly selected with replacement, find the probability that they both were killed.

Short answer

The probability that two randomly selected drivers chosen with replacement are killed is 0.198.

Step-by-step solution

Given information

The drivers of passenger cars are categorized into four categories.

 Step 2: Define multiplication rule

The chances of joint occurrences of events are measured as the product of an event’s individual probability. Two types of scenarios are possible:

Independent events: Involve the selection of events with replacement

Dependent events: Involve the selection of events without replacement

Independent events have the same probabilities irrespective of the sequence of selection.

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

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

Compute the probability using the multiplication rule

Define the events as follows:

E: The randomly selected driver was killed

A: The two drivers randomly selected with replacement are killed

The number of drivers who were killed is 8057.

The total number of drivers surveyed is 18102.

Thus, the probability that any randomly selected driver was killed is:

$P\left(E\right)=\frac{8057}{18102}$

As the selection is made with replacement, the probability that the second driver was killed remains the same since the events are independent.

The probability that two drivers randomly selected with replacement are killed is:

$$\begin{gathered} P\left(E\text{and}F\right)=P\left(E\right)\times P\left(F\right) \\ =\frac{8057}{18102}\times \frac{8057}{18102} \\ =0.198 \end{gathered}$$

Thus, the probability that two drivers randomly selected with replacement are killed is 0.198.