A certain type of dvd player is sold at just two stores. 30% of sales are from store A and 70% of sales are from store B. 2.7% of the dvd players sold at store A are defective while 3.7% of the dvd players sold at store B are defective. If Kate receives one of these dvd players as a gift and finds that it is defective, what is the probability that is came from store A?

Events:

A=from store A
B=from store B
D=defective
then probability of defective player given it is from store A:
P(D|A)=P(D∩A)/P(A)=P(D∩A)/0.3=0.027
then probability of defective player given it is from store B:
Solve for P(D∩A)
P(D|B)=P(D∩B)/P(B)=P(D∩B)/0.7=0.037
Solve for P(D∩B)

Probability that a player is defective from any store
=P(D)
=P(D∩(A∪B))
=P(D∩A)+P(D∩B)
=sum of two values calculated above

Probability that a player is from store A, given that it is defective
P(A|D)
=P(A∩D)/P(D)
=P(D∩A)/P(D)
= evaluate using known values calculated above.
(hint: the answer is between 0.2 and 0.25)

To find the probability that the defective DVD player came from Store A, we can use Bayes' theorem.

Let's assign the following probabilities:
A: The DVD player came from Store A
B: The DVD player came from Store B
D: The DVD player is defective

We want to find P(A|D), the probability that the DVD player came from Store A given that it is defective, which can be calculated using Bayes' theorem:

P(A|D) = (P(D|A) * P(A)) / P(D)

The information given in the problem statement is as follows:
P(A) = 0.30 (30% of sales are from Store A)
P(B) = 0.70 (70% of sales are from Store B)
P(D|A) = 0.027 (2.7% of DVD players sold at Store A are defective)
P(D|B) = 0.037 (3.7% of DVD players sold at Store B are defective)

To calculate P(D), we need to use the law of total probability.

P(D) = P(D|A) * P(A) + P(D|B) * P(B)
= 0.027 * 0.30 + 0.037 * 0.70

Now we can substitute these values into Bayes' theorem:

P(A|D) = (P(D|A) * P(A)) / P(D)
= (0.027 * 0.30) / (0.027 * 0.30 + 0.037 * 0.70)

Calculating this expression will give us the probability that the defective DVD player came from Store A.