Suppose that 70 of drivers are careful and 30 are reckless S

Suppose that 70% of drivers are \"careful\" and 30% are \"reckless.\" Suppose further that a careful driver has a 0.1 probability of being in an accident in a given year, while for a reckless driver the probability is 0.4. What is the probability that a randomly selected driver will have an accident within a year?

Solution

Let C be the event that a randomly selected driver is \"careful\"

Let R be the event that a randomly selected driver is \"reckless\"

Let A be the event that a randomly selected driver is met with accident

given that P(C)=0.70 and P( R)=0.30 and the conditional probabilities P(A/C) =0.10 and P(A/R)=0.40

To find the Probability that a randomly selected driver will have an accident =P(A)=P(A/C).P(C)+P(A/R).P(R)

(Using the Theorem on Total Probability)

required probability = P(A)=(0.10 )(0.70)+ ( 0.40)(0.30)=0.12=12/100 = 12%