Teenage drivers pay more for automobile insurance than older drivers. Also, many insurance companies offer discounts for teenage drivers that achieve good grades. Assume that 20% of all teenage drivers are involved in accidents each year. Assume that 10% of all teenage drivers have GPAs that are 3.5 or above. Of the teenage drivers involved in accidents, only 5% of them have GPAs that are 3.5 or above.

a) Let A be the event that a teenager has a GPA of 3.5 or above. Let B be the event that a teenage driver is involved in an accident this year. Find P(A) and P(B).
b) Find P(A given B)
c) Find the probability that a randomly chosen teenage driver has a GPA of 3.5 or above and is involved in an accident this year. (5 points)

a) To find the probabilities, we can use the given information.

P(A) represents the probability of a teenage driver having a GPA of 3.5 or above. From the information given, we know that 10% of all teenage drivers have GPAs that are 3.5 or above. Therefore, P(A) = 0.10 or 10%.

P(B) represents the probability of a teenage driver being involved in an accident this year. The information states that 20% of all teenage drivers are involved in accidents each year. Therefore, P(B) = 0.20 or 20%.

b) P(A given B) refers to the probability of a teenage driver having a GPA of 3.5 or above, given that they are involved in an accident. To find this probability, we need to use the formula for conditional probability: P(A | B) = P(A ∩ B) / P(B).

From the information given, we know that only 5% of teenage drivers involved in accidents have GPAs that are 3.5 or above. Therefore, P(A ∩ B) = 0.05 or 5%.

Using the value calculated in part a, P(B) = 0.20 or 20%, we can substitute these values into the formula to find P(A given B):
P(A | B) = P(A ∩ B) / P(B) = 0.05 / 0.20 = 0.25 or 25%.

c) To find the probability that a randomly chosen teenage driver has a GPA of 3.5 or above and is involved in an accident this year, we can use the formula for joint probability: P(A ∩ B) = P(A) * P(B), where P(A ∩ B) represents the probability that both events A and B occur.

Using the values calculated in part a, P(A) = 0.10 or 10% and P(B) = 0.20 or 20%, we can substitute these values into the formula to find P(A ∩ B):
P(A ∩ B) = P(A) * P(B) = 0.10 * 0.20 = 0.02 or 2%.

Therefore, the probability that a randomly chosen teenage driver has a GPA of 3.5 or above and is involved in an accident this year is 2%.