Among 26 to 31 year olds 35% say they have driven a car while under the influence of peer pressure. suppose 2 26 to 31 year olds are selected at random.
A)the probability that at least 1 has not driven under the influence?
B)the probability that at least 1 of the 2 adults has driven under the influence?
To calculate the probability in this scenario, we need to use the concept of independent events and the complement rule. Let's break down each question step by step:
A) The probability that at least 1 person has not driven under the influence:
To find this probability, we need to calculate the complement of the event "both individuals have driven under the influence." We can then subtract this value from 1 to get the probability of at least one person not having driven under the influence.
1. Calculate the probability that both individuals have driven under the influence:
- The probability of one person having driven under the influence is 35% (or 0.35).
- Since the events are independent, the probability of both individuals having driven under the influence is the product of their individual probabilities: 0.35 * 0.35 = 0.1225.
2. Calculate the complement of this event:
- The complement of both individuals having driven under the influence is that at least one person has not driven under the influence.
3. Subtract the complement from 1 to find the probability of at least 1 person not having driven under the influence:
- Probability(at least 1 has not driven) = 1 - Probability(both have driven) = 1 - 0.1225 = 0.8775.
Therefore, the probability that at least one person has not driven under the influence is 0.8775 (or 87.75%).
B) The probability that at least 1 of the 2 adults has driven under the influence:
In this case, we want to find the probability that at least one person has driven under the influence. This is the complement of the event "both individuals have not driven under the influence."
1. Calculate the probability that both individuals have not driven under the influence:
- The probability of one person not having driven under the influence is 1 - 0.35 = 0.65.
- Since the events are independent, the probability of both individuals not having driven under the influence is the product of their individual probabilities: 0.65 * 0.65 = 0.4225.
2. Calculate the complement of this event:
- The complement of both individuals not having driven under the influence is that at least one person has driven under the influence.
3. Subtract the complement from 1 to find the probability of at least 1 person having driven under the influence:
- Probability(at least 1 has driven) = 1 - Probability(both have not driven) = 1 - 0.4225 = 0.5775.
Therefore, the probability that at least one person has driven under the influence is 0.5775 (or 57.75%).