A company insures red and black cars, male and female drivers and writes policies in 2 territories (A and B). There are 300 male drivers and 200 female drivers in total. There are 150 males who drive red cars and 100 females who drive red cars. 100 male and 100 female drivers live in territory A and 50 of each, males and females, drive red cars in territory A. Given that a randomly selected policyholder drives a black car, what is the probability that they are female and live in territory B?

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TTanishq

Explanation:

Explanation

We need to find the conditional probability: $P(\text{Female and Territory B} \mid \text{Black Car})$

From the given information:

Total drivers: 300 male + 200 female = 500 drivers
Male drivers: 300 total
- Male red car: 150
- Male black car: 300 - 150 = 150
Female drivers: 200 total
- Female red car: 100
- Female black car: 200 - 100 = 100

Territory A breakdown:

Male drivers in A: 100
- Male red car in A: 50
- Male black car in A: 100 - 50 = 50
Female drivers in A: 100
- Female red car in A: 50
- Female black car in A: 100 - 50 = 50

Territory B breakdown:

Male drivers in B: 300 - 100 = 200
- Male red car in B: 150 - 50 = 100
- Male black car in B: 200 - 100 = 100
Female drivers in B: 200 - 100 = 100
- Female red car in B: 100 - 50 = 50
- Female black car in B: 100 - 50 = 50

$P(\text{Female and Territory B} \mid \text{Black Car}) = \frac{P(\text{Female, Territory B and Black Car})}{P(\text{Black Car})}$

Numerator: Female black car in territory B = 50

Denominator: Total black cars = Male black cars + Female black cars = 150 + 100 = 250

$P = \frac{50}{250} = 0.20 \text{ or } 20\%$

Therefore, the probability that a randomly selected black car driver is female and lives in territory B is 20%.

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