Question

use the data in the following table, which lists survey results from high school drivers at least 16 years of age. assume that subjects are randomly selected from those include table. if four different high school drivers are randomly selected, find the probability that they all drove when drinking alcohol. the probability that four randomly selected high school drivers all drove when drinking alcohol is 0.000136. (round to six decimal places as needed.)

Explanation:

Step1: Calculate the total number of high - school drivers

We sum up all the values in the table:

$$684 + 2873+192 + 4339=8088$$

Step2: Calculate the number of drivers who drove when drinking alcohol

We sum up the values in the "Yes" column:

$$684+192 = 876$$

Step3: Calculate the probability of the first selection

The probability \(P_1\) that the first - selected driver drove when drinking alcohol is \(\frac{876}{8088}\)

Step4: Calculate the probability of the second selection

After the first selection (without replacement), there are \(n = 8087\) drivers left and \(m=875\) drivers who drove when drinking alcohol left. So the probability \(P_2=\frac{875}{8087}\)

Step5: Calculate the probability of the third selection

After the second selection (without replacement), there are \(n = 8086\) drivers left and \(m = 874\) drivers who drove when drinking alcohol left. So the probability \(P_3=\frac{874}{8086}\)

Step6: Calculate the probability of the fourth selection

After the third selection (without replacement), there are \(n = 8085\) drivers left and \(m = 873\) drivers who drove when drinking alcohol left. So the probability \(P_4=\frac{873}{8085}\)

Step7: Calculate the combined probability

By the multiplication rule for dependent events \(P = P_1\times P_2\times P_3\times P_4\)

$$P=\frac{876}{8088}\times\frac{875}{8087}\times\frac{874}{8086}\times\frac{873}{8085}$$

$$P=\frac{876\times875\times874\times873}{8088\times8087\times8086\times8085}$$

$$876\times875\times874\times873=(876\times875)\times(874\times873)=766500\times762002 = 584344533000$$

$$8088\times8087\times8086\times8085=(8088\times8087)\times(8086\times8085)=65503656\times65503610=4290797790100160$$

$$P=\frac{584344533000}{4290797790100160}\approx0.000136$$

Answer:

\(0.000136\)