Use Bayes' theorem to solve this problem. A storeowner purchases stereos from two companies. From Company A, 550 stereos are purchased and 1% are found to be defective. From Company B, 850 stereos are purchased and 6% are found to be defective. Given that a stereo is defective, find the probability that it came from Company A.
I plugged in the numbers and here's my answer...
Company A: 1% x 550 = 5.5 are bad
Company B: 6% x 850 = 51 are bad
Total bad of defective stereos = 5.5 + 51 = 56.5
P(A | Bad) = 5.5 / 56.5 = 11/113
So Probability that the defective one is from A = 0.097345
Baye's theorem
P(P|Q)=P(P∩Q)/P(Q)=P(Q∩P)/P(Q)
=P(Q∩P)/P(P)*(P(P)/P(Q))
=P(Q|P)*(P(P)/P(Q))
or simply
P(P|Q)=P(Q|P)*(P(P)/P(Q))
Here
P=stereo came from company A
Q=stereo is defective
So
P(P|Q) probability that the stereo came from company A given that it is defective.
P(Q|P) probability that the stereo is defective given that it came from company A. (this value is given).
P(P)=probability that the stereo came from company A = 550/total no. bought
P(Q)=probability that a stereo is defective
=total number of defectives / total no. of stereos bought.
Plug in the above numbers to get the answer. Post for a check if you wish.
Well, let's put on our detective hats and solve this mystery using Bayes' theorem, shall we?
First, let's define our events:
A: Stereo is from Company A
B: Stereo is from Company B
D: Stereo is defective
We want to find the probability that a stereo came from Company A given that it is defective:
P(A|D)
According to Bayes' theorem:
P(A|D) = (P(D|A) * P(A)) / P(D)
Now, let's break down the given information:
P(D|A) = 0.01 (1% of stereos from Company A are defective)
P(D|B) = 0.06 (6% of stereos from Company B are defective)
P(A) = 550/(550 + 850) = 0.393 (probability of selecting a stereo from Company A)
P(B) = 1 - P(A) = 0.607 (probability of selecting a stereo from Company B)
P(D) can be calculated using the Law of Total Probability:
P(D) = P(D|A) * P(A) + P(D|B) * P(B)
Plugging in the values, we get:
P(D) = (0.01 * 0.393) + (0.06 * 0.607)
P(D) = 0.00393 + 0.03642
P(D) = 0.04035
Now, we can calculate P(A|D):
P(A|D) = (P(D|A) * P(A)) / P(D)
P(A|D) = (0.01 * 0.393) / 0.04035
P(A|D) = 0.00393 / 0.04035
P(A|D) ≈ 0.0973
So, the probability that a defective stereo came from Company A is approximately 0.0973, or about 9.73%. Now, go ahead and tell the storeowner to keep an eye on the clowns at Company A!
To solve this problem using Bayes' theorem, we need to calculate the probability that a stereo came from Company A given that it is defective. Let's denote the events as follows:
A: The stereo comes from Company A
B: The stereo comes from Company B
D: The stereo is defective
We need to find P(A|D), i.e., the probability that the stereo comes from Company A given that it is defective.
According to Bayes' theorem, we have:
P(A|D) = (P(D|A) * P(A)) / P(D)
P(D|A) = 1% = 0.01 (probability of a stereo being defective given it comes from Company A)
P(A) = 550 / (550 + 850) = 550 / 1400 = 0.393 (probability of selecting a stereo from Company A)
P(D) = (P(D|A) * P(A)) + (P(D|B) * P(B))
= (0.01 * 0.393) + (0.06 * 0.607)
= 0.00393 + 0.03642
= 0.04035 (probability of selecting a defective stereo)
Now, we can substitute these values into the Bayes' theorem equation:
P(A|D) = (0.01 * 0.393) / 0.04035
= 0.00393 / 0.04035
≈ 0.09725
Therefore, the probability that a defective stereo came from Company A is approximately 0.09725 (or 9.725%).
To solve this problem using Bayes' theorem, we need to calculate the probability that a stereo came from Company A given that it is defective.
Let's define the following events:
A = The stereo came from Company A
B = The stereo came from Company B
D = The stereo is defective
We are given the following probabilities:
P(A) = Probability of choosing a stereo from Company A = 550 / (550 + 850) = 0.393
P(B) = Probability of choosing a stereo from Company B = 850 / (550 + 850) = 0.607
P(D|A) = Probability that a stereo is defective given it came from Company A = 0.01
P(D|B) = Probability that a stereo is defective given it came from Company B = 0.06
We need to find P(A|D), which is the probability that a stereo came from Company A given that it is defective. Applying Bayes' theorem, we can write:
P(A|D) = (P(D|A) * P(A)) / P(D)
To calculate P(D), which is the probability of a stereo being defective, we can use the law of total probability:
P(D) = P(D|A) * P(A) + P(D|B) * P(B)
Substituting the given values:
P(D) = (0.01 * 0.393) + (0.06 * 0.607)
Simplifying, we find:
P(D) = 0.00393 + 0.03642
P(D) = 0.04035
Now, substituting the values back into Bayes' theorem:
P(A|D) = (0.01 * 0.393) / 0.04035
Simplifying further, we find:
P(A|D) = 0.00393 / 0.04035
P(A|D) ≈ 0.0974
Therefore, the probability that a defective stereo came from Company A is approximately 0.0974, or 9.74%.