Question

The government is performing a study on the use of green bags when grocery shopping. It was found that a customer will shop at Coles with probability 0.39, Woolworths with probability 0.14, and Aldi with probability 0.47. A customer has a probability of 0.53, 0.21, 0.46 of using green bags if they shop at Coles, Woolworths, or Aldi (respectively).
What is the probability that a random customer will use green bags?



What is the probability that a person who uses green bags shops at Aldi?



What is the probability that a Coles customer does not use green bags?



What is the probability that a person shops at Coles and does not use green bags?

Answer 1

Solution :

Let us define some events as follows :

A : A customer shops at Coles

B : A customer shops at Woolworths

C : A customer shops at Aldi

D : A customer uses green bag

We have following informations :

P(A) = 0.39, P(B) = 0.14, P(C) = 0.47

P(D | A) = 0.53, P(D | B) = 0.21 and P(D | C) = 0.46

a) We have to find P(D).

A customer uses green bags then it means either he shops at Coles and uses green bags or he shops at Woolworths and uses green bags or he shops at Aldi and uses green bags.

i.e. P(D) = P(A ∩ D) + P(B ∩ D) + P(C ∩ D)

Now,

P(A ∩ D) = P(A | D) × P(A)

P(A ∩ D) = (0.53 × 0.39) = 0.2067

P(B ∩ D) = P(B | D) × P(B)

P(B ∩ D) = (0.21 × 0.14) = 0.0294

P(C ∩ D) = P(C | D) × P(C)

P(C ∩ D) = (0.46 × 0.47) = 0.2162

Hence, P(D) = 0.2067 + 0.0294 + 0.2162

P(D) = 0.4523

Hence, the probability that a random customer will use green bags is 0.4523.

b) We have to find P(C | D)

From part (a) we have, P(C ∩ D) = 0.2162 and P(D) = 0.4523

Hence, the probability that a person who uses green bags shops at Aldi is 0.4780.

c) We have to find .

We have, P(D | A) = 0.53

Hence, the probability that a Coles customer does not use green bags is 0.47.

d) We have to find .

From part (a) we have, P(A) = 0.39 and P(A ∩ D) = 0.2067

Hence, the probability that a person shops at Coles and does not use green bags is 0.1833.