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

Seatbelt Use or Driver Not Killed If one of the drivers is randomly selected, find the probability of getting a driver who used a seatbelt or was killed.

Short answer

The probability that a randomly selected driver was killed or used a seatbelt is 0.832.

Step-by-step solution

Given information

The table contains data concerning drivers of passenger cars involved in a head-on collision.

Step 2:Describe conditional probability

The idea of conditional probability is to compute an event’s probability relative to a prior event. It is computed as follows:

$P\left(A\left|B\right.\right)=\frac{P\left(A\text{and}B\right)}{P\left(B\right)}$

Describe the addition rule

The addition rule is used to compute the probability of occurrence of event A or B stated as follows:

$P\left(A\text{or}B\right)=P\left(A\right)+P\left(B\right)-P\left(A\text{and}B\right)$

Calculate the totals 

Tabulate the sum of rows and columns as follows:

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

Compute the probability 

Define the events as follows:

E: The randomly selected driver used a seatbelt.

F: The randomly selected driver was killed.

The number of drivers who used seatbelts is 10660.

The number of drivers killed is 8057.

The number of drivers who were killed and used seatbelts is 3655.

The total number of drivers surveyed is 18102.

The probabilities are computed as:

$$\begin{gathered} P\left(E\right)=\frac{10660}{18102} \\ P\left(F\right)=\frac{8057}{18102} \\ P\left(E\text{and}F\right)=\frac{3655}{18102} \end{gathered}$$

The probability that a randomly selected driver was killed or used or seatbelts is:

$$\begin{gathered} P\left(E\text{or}F\right)=P\left(E\right)+P\left(F\right)-P\left(E\text{and}F\right) \\ =\frac{10660}{18102}+\frac{8057}{18102}-\frac{3655}{18102} \\ =0.832 \end{gathered}$$

Thus, the probability that a randomly selected driver was killed or used seatbelts is 0.832.