Question

The probability a household in a community uses gas for cooking is 0.18. If a shopper from the community is from a household that uses gas for cooking, there is a 0.7 probability the person shops at Prime Foods. If a shopper is from a household that does not use gas for cooking, there is a 0.35 probability the person shops at Prime Foods. If a person from the community does not shop at Prime Foods, what is the probability gas is used for cooking at that household

Answer 1

The probability of a person from the community not shopping at Prime Foods given that he uses gas for cooking at his household is 0.092

Explanation:

Hello!

Given the information:

A: "The household uses gas for cooking" ⇒ P(A)= 0.18

B: "The person from the comunity shops at Prime Foods"

The person shops at Prime Foods given that he uses gas for cooking:

P(B/A)= 0.7

The person shops at Prime Foods given that he doesn't use gas for cooking:

P(B/Ac)= 0.35

If the person doesn't shop at Prime Foods, what is the probability that he uses gas for cooking?

Symbolically:

P(A/Bc) = P(A∩Bc)

P(Bc)

The event Bc is the complement of event B, so it's probability is P(Bc)= 1 - P(B)

You can calculate the probability of B as:

P(B)= P(A∩B)+ P(Ac∩B)

Using the information of P(B/A)= 0.7 and P(B/Ac)= 0.35 you can reach the values of both intersections.

P(B/A)= P(A∩B)

P(A)

P(B/A)*P(A)= P(A∩B)

P(A∩B)= 0.7*0.18= 0.126

and

P(B/Ac)= P(Ac∩B)

P(Ac)

If A and Ac are complementary events, then P(Ac)= 1 - P(A)= 1 - 0.18= 0.82

Then:

P(B/Ac)*P(Ac)= P(Ac∩B)

P(Ac∩B)= 0.35*0.82= 0.287

The probability of B is:

P(B)= P(A∩B)+ P(Ac∩B)= 0.126+0.287= 0.413

P(Bc)= 1 - P(B)= 1 - 0.413= 0.587

And

P(A)= P(A∩B)+P(A∩Bc)

P(A∩Bc)=P(A)=-(A∩B)= 0.18-0.126= 0.054

Finally:

P(A/Bc) = P(A∩Bc) =0.054 = 0.0919≅ 0.092

P(Bc) 0.587

I hope it helps!