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

No Seatbelt or Driver Not Killed If one of the drivers is randomly selected, find the probability of getting someone who did not use a seatbelt or was not killed

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

The addition rule is defined as:

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

It computes the probability of occurrence of event A or B using the probability of A, B and their joint occurrence.

Add all values in the rows and columns and note under totals.

Define the events as follows:

E: The randomly selected driver did not use a seatbelt.

F: The randomly selected driver was not killed.

The number of drivers whodid not use seatbelts is 7442.

The number of drivers who were notkilled is 10045.

The number of drivers who were neither killed nor used seatbelts is 3040.

The total number of drivers surveyed is 18102.

The probabilities are computed as:

$$\begin{gathered} P\left(E\right)=\frac{7442}{18102} \\ P\left(F\right)=\frac{10045}{18102} \\ P\left(E\text{and}F\right)=\frac{3040}{18102} \end{gathered}$$

The probability that a randomly selected driver was neither killed nor used a seatbelt is:

$$\begin{gathered} P\left(E\text{or}F\right)=P\left(E\right)+P\left(F\right)-P\left(E\text{and}F\right) \\ =\frac{7442}{18102}+\frac{10045}{18102}-\frac{3040}{18102} \\ =0.798 \end{gathered}$$

Thus, the probability that a randomly selected driver was neither killed nor used seatbelts is 0.798.