Question

In a certain​ state, about 70​% of drivers who are arrested for driving while intoxicated​ (DWI) are convicted.

a. If 12 independently selected drivers were arrested for​ DWI, how many of them would you expect to be​ convicted? Choose the correct answer below.

A. 6

B. 9

C. 7

D. 8

Your answer is correct. b.

What is the probability that exactly 8 out of 12 independently selected drivers are​ convicted?

The probability that exactly 8 out of 12 12 drivers are convicted is . 231.

​(Type an integer or decimal rounded to three decimal places as​ needed.)

c. What is the probability that 8 or fewer are​ convicted?

The probability that 8 or fewer drivers are convicted is ________.

​(Type an integer or decimal rounded to three decimal places as​ needed.)

Answer 1

Binomial Distribution :

Binomial Distribution

If 'X' is the random variable representing the number of successes, the probability of getting ‘r’ successes and ‘n-r’ failures, in 'n' trails, ‘p’ probability of success ‘q’=(1-p) is given by the probability function

Expected Value of X : E(X) = np

For the Given problem,

Probability of a driver arrested for driving while intoxicated​ (DWI) are convicted : p= 70/100= 0.7

q = 1-p = 1-0.7 = 0.3

Number of  independently selected drivers were arrested for​ DWI = 12

X : Number of  selected drivers that  are convicted

X follows a Binomial Distribution with probability mass function

a. If 12 independently selected drivers were arrested for​ DWI, how many of them would you expect to be​ convicted

i.e E(X)

E(X) = np = 12 x 0.7 = 8.4

b.

probability that exactly 8 out of 12 independently selected drivers are​ convicted = P(X=8)

The probability that exactly 8 out of 12 drivers are convicted is . 231.

c. probability that 8 or fewer are​ convicted = P(X8)

P(X8) = 1 - P(X>8)

P(X>8) = P(X=9)+P(X=10)+P(X=11)+P(X=12)

P(X>8) = P(X=9)+P(X=10)+P(X=11)+P(X=12) = 0.4925

P(X8) = 1 - P(X>8) = 1-0.4925=0.5075

The probability that 8 or fewer drivers are convicted is 0.5075