Question

2. Safety Insurance Company has compiled the following statistics. Twenty percent of policyholders are males under 25 years of age. Forty percent of policyholders are males 25 and over. Ten percent of policyholders are females under 25 years of age. Thirty percent of policyholders are females 25 and over. For any one year period, twenty-two percent of males under 25 years of age report an accident. Fifteen percent of males 25 and older report an accident. Sixteen percent of females under 25 report accidents. Fourteen percent of females 25 and older report accidents. Given that an accident is reported what is the probability that it came from:

a. males under 25

b. males 25 and over

c. females under 25

d. females 25 and over

*** Please show work

Answer 1

We are given here that:
P( males < 25) = 0.2
P( males >= 25) = 0.4
P( females < 25) = 0.1
P( females >= 25) = 0.3

Also, we are given here that:
P( accident | males < 25) = 0.22
P( accident | males >= 25) = 0.15
P( accident | females < 25) = 0.16
P( accident | females >= 25) = 0.14

Using law of total probability, we get here:
P( accident ) = P( accident | males < 25)P( males < 25) + P( accident | males >= 25)P( males >= 25) + P( accident | females < 25) P( females < 25) + P( accident | females >= 25)P( females >= 25)

P( accident) = 0.22*0.2 + 0.15*0.4 + 0.16*0.1 + 0.14*0.3 = 0.162

Now using Bayes theorem, we get the required probabilities now as:

a) P( males < 25 | accident) = P( accident | males < 25)P( males < 25) / P( accident )

= 0.22*0.2 / 0.162

= 0.2716

Therefore 0.2716 is the required probability here.

b) P( males >= 25 | accident) = P( accident | males >= 25)P( males >= 25) / P( accident )

= 0.15*0.4 / 0.162

= 0.3704

Therefore 0.3704 is the required probability here.

c) P( females < 25 | accident) = P( accident | females < 25)P( females < 25) / P( accident )

= 0.16*0.1 / 0.162

= 0.0988

Therefore 0.0988 is the required probability here.

d) P( females >= 25 | accident) = P( accident | females >= 25)P( females >= 25) / P( accident )

= 0.14*0.3 / 0.162

= 0.2593

Therefore 0.2593 is the required probability here.