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

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

The formula for computing conditional probability is stated below.

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

Here, the $P\left(A|B\right)$ is defined as the probability of an event A, given that event Boccurred in the past,computed as the ratio of probability for the occurrence of both A and B over the probability of event B.

The additive totals are:

Let E be the event that a driver was killed and F be the event that the driver did not use a seatbelt.

The number of drivers who were not using a seatbelt is 7442.

The number of drivers killed is 8057.

The number of drivers who did not usea seatbelt and were killed is 4402.

The total number of drivers surveyed is 18102.

Using the values, you get:

$$\begin{gathered} P\left(E\text{and}F\right)=\frac{4402}{18102} \\ P\left(F\right)=\frac{7442}{18102} \end{gathered}$$

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

$$\begin{gathered} P\left(E\left|F\right.\right)=\frac{P\left(E\text{and}F\right)}{P\left(F\right)} \\ =\frac{\frac{4402}{18102}}{\frac{7442}{18102}} \\ =\frac{4402}{7442} \\ =0.592 \end{gathered}$$

Thus, the probability that a randomly selected driver was killed, given that he did not use a seatbelt, is 0.592.