A company has four production sections viz., S1, S2, S3 and S4 which contribute 30%, 20%, 28% and 22% of the total output. It was observed that those sections respectively produced 1%, 2%, 3% and 4% defective units. If a unit is selected at random and found to be defective, what is the probability that the unit so selected has come from either S1 or S4.
Hint: We know
P(S1)=0.30, P(S2)=0.20, P(S3)=0.28, P(S4)=0.22. If D is the defect case than we know also P(D|S1)=0.01, P(D|S2)=0.02, P(D|S3)=0.03,and P(D|S4)=0.04.
Now, the question is asking you to determine P(S1|D)+P(S4|D). Can you determine these probabilities with the Bayes' theorem?
ariyilla
0.59
To find the probability that the defective unit has come from either S1 or S4, we need to calculate the conditional probability.
Let's denote the event of selecting a defective unit as D, and the event of selecting a unit from S1 or S4 as E.
We are given that the probability of selecting a defective unit in S1 is 1% and in S4 is 4%.
Now, we can calculate the probability of selecting a defective unit from either S1 or S4.
P(D|E) = (P(D AND E)) / P(E)
P(D AND E) is the probability of selecting a defective unit from either S1 or S4, and P(E) is the probability of selecting a unit from S1 or S4.
To calculate P(D AND E), we need to sum up the probabilities of selecting a defective unit from S1 and S4:
P(D AND E) = P(D in S1) + P(D in S4)
= 1% + 4%
= 5%
To calculate P(E), we need to sum up the probabilities of selecting a unit from S1 and S4:
P(E) = P(S1) + P(S4)
= 30% + 22%
= 52%
Now, we can calculate P(D|E) using the formula:
P(D|E) = P(D AND E) / P(E)
= 5% / 52%
≈ 0.09615
So, the probability that the unit selected, which is defective, has come from either S1 or S4 (E) is approximately 0.09615, or 9.615%.