Question

The table shows the outcome of car accidents in a certain state for a recent year by whether or not the driver wore a seat belt. Find the probability of wearing a seat belt, given that the driver did not survive a car accident. Part 1: The probability as a decimal is _ (Round to 3 decimal places as needed.) Part 2: The probability as a fraction is _

Answer 1

Conditional probability is a measure of the probability of an event occurring, given that another event has already occurred.

The table shows the outcome of car accidents by whether or not the driver wearing a seat belt.

Let's call:

A = The event of the driver wearing a seat belt in a car accident.

B = The event of the driver dying in a car accident

The conditional probability is calculated as follows:

`P(A|B)=(P(A\cap B))/(P(B))`

The conditional probability stated in the formula is that for the driver wearing a seat belt knowing he did not survive the car accident.

The numerator of the formula is the probability of both events occurring, i.e., the driver wore a seat belt and died. The denominator is the simple probability that the driver died in a car accident.

From the table, we can intersect the first column and the second row to find the number of outcomes where both events occurred. The probability of A ∩ B is:

`P(A\cap B)=(511)/(583,470)`

The probability of B is:

`P(B)=(2217)/(583,470)`

The required probability is:

`P(A|B)=((511)/(583,470))/((2217)/(583,470))`

Simplifying the common denominators:

`P(A|B)=(511)/(2217)=0.230`