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

All Three Successful If 3 drivers are randomly selected without replacement, find the probability that none of them were killed.

Short answer

The probability that none of the three randomly selected drivers are killed when selected without replacement is 0.171.

Step-by-step solution

Given information

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

Define multiplication rule

Consider n events that occur together be defined as $A_1,A_2,...,A{}_n$which occur simultaneously. Then, the probability of the joint of occurrence is stated below.

$P\left(A_{1}and{A}_{2}and...and{A}_n\right)=P\left(A_1\right)\times P\left(A_2\right)\times ...\times P\left(A_n\right)$

In case the events occur without replacement, the probability of successive events changes.

Step 3:Compute the probability of individual events

The additive total for each row or column is:

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

Define the events as follows:

A: Event of randomly selecting the first driver who was killed

B: Event of randomly selecting the second driver who was killed

C: Event of randomly selecting the third driver who was killed

The number of drivers who were killed is 10045.

The total number of drivers who were killed is 18102.

The probability of selecting the first driver who was not killed is:

$P\left(A\right)=\frac{10045}{18102}$

The number of drivers who were killed, after selecting the first one, is 10044.

The total number of drivers who were killed, after selecting the first one, is 18101.

The probability of selecting the second driver who was not killed is:

$P\left(B\right)=\frac{10044}{18101}$

The number of drivers who were killed, after selecting the second one, is 10043.

The total number of drivers who were killed, after selecting the first one, is 18100.

The probability of selecting the third driver who was not killed is:

$P\left(C\right)=\frac{10043}{18100}$

Compute the overall probability 

Using the multiplication rule, the probability that none of the drivers was killed when selected without replacement is:

$$\begin{gathered} P\left(A\text{and}B\text{and}C\right)=\frac{10045}{18102}\times \frac{10044}{18101}\times \frac{10043}{18100} \\ =0.1708 \\ \approx 0.171 \end{gathered}$$

Thus, the probability that none of the three drivers was killed when selected without replacement is 0.171.