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 Find the probability of randomly selecting a driver and getting one who was not killed given that the driver was using a seatbelt.

Short answer

The probability that the randomly selected driver was not killed, given that he uses a seatbelt, is approximately 0.657.

Step-by-step solution

Given information

The data for passenger cars involved in a head-on collision is provided.

Step 2:Define the concept of probability

The formula for the probability of any arbitrary event A is defined as follows:

$P\left(A\right)=\frac{\text{Number}\text{of}\text{favorable}\text{outcomes}}{\text{Total}\text{number}\text{of}\text{outcomes}}$

In the case of conditional probability, the probability of an event A, given that event B has taken place, is the ratio of two values, probability for both A and B, over the probability of event B.

Mathematically,

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

Step 3:Compute the totals row-wise and column-wise

Add the values in rows and columns to obtain the 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 

Define F as the event that a randomly selected driver uses a seatbelt.

Define E as the event that a randomly selected driver was not killed.

The favorable number of drivers who use a seatbelt is 10660.

The favorable number of drivers who are not killed is 10045.

The favorable number of drivers who use a seatbelt and were not killed is 7005.

The total number of drivers surveyed is 18102.

Using the values, you get:

$$\begin{gathered} P\left(E\text{and}F\right)=\frac{7005}{18102} \\ P\left(F\right)=\frac{10660}{18102} \end{gathered}$$

The probability that a randomly selected driver was not killed, given that he uses a seatbelt, is:

$$\begin{gathered} P\left(E\left|F\right.\right)=\frac{P\left(EandF\right)}{P\left(F\right)} \\ =\frac{\frac{7005}{18102}}{\frac{10660}{18102}} \\ =\frac{7005}{10660} \\ =0.657 \end{gathered}$$

Thus, the probability that the randomly selected driver was not killed, given that he uses a seatbelt, is 0.657.