Questions: Among 53 - to 58 -year-olds, 37% say they have driven a car while under the influence of alcohol. Suppose three 53- to 58-year-olds are selected at random. Complete parts (a) through (d) below. influence of alcohol? 0.0507 (Round to four decimal places as needed.) (b) What is the probability that at least one has not driven a car while under the influence of alcohol? 0.9493 (Round to four decimal places as needed.) (c) What is the probability that none of the three have driven a car while under the influence of alcohol? 0.2500 (Round to four decimal places as needed.) (d) What is the probability that at least one has driven a car while under the influence of alcohol? (Round to four decimal places as needed.)

Solution Steps

To solve these probability questions, we can use the concept of complementary probability and the binomial probability formula. Here's how to approach each part:

(a) Calculate the probability that all three have driven under the influence by raising the probability of one person driving under the influence to the power of three.

(b) Use the complementary probability: the probability that at least one has not driven under the influence is 1 minus the probability that all three have driven under the influence.

(c) Calculate the probability that none have driven under the influence by raising the probability of one person not driving under the influence to the power of three.

(d) Use the complementary probability: the probability that at least one has driven under the influence is 1 minus the probability that none have driven under the influence.

To find the probability that all three individuals have driven under the influence, we use the formula:

\[ P(\text{all driven}) = (0.37)^3 \]

Calculating this gives:

\[ P(\text{all driven}) = 0.050653 \]

Rounded to four decimal places, this is:

\[ P(\text{all driven}) = 0.0507 \]

The probability that at least one has not driven under the influence is the complement of all three having driven:

\[ P(\text{at least one not driven}) = 1 - P(\text{all driven}) \]

\[ P(\text{at least one not driven}) = 1 - 0.050653 = 0.949347 \]

Rounded to four decimal places, this is:

\[ P(\text{at least one not driven}) = 0.9493 \]

To find the probability that none have driven under the influence, we use:

\[ P(\text{none driven}) = (1 - 0.37)^3 \]

Calculating this gives:

\[ P(\text{none driven}) = 0.250047 \]

Rounded to four decimal places, this is:

\[ P(\text{none driven}) = 0.2500 \]

The probability that at least one has driven is the complement of none having driven:

\[ P(\text{at least one driven}) = 1 - P(\text{none driven}) \]

\[ P(\text{at least one driven}) = 1 - 0.250047 = 0.749953 \]

Rounded to four decimal places, this is:

\[ P(\text{at least one driven}) = 0.7500 \]

Final Answer