Question

A lot of 1000 components contains 350 that are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective, and let B be the event that the second component drawn is defective.
a. Find P(A).b. Find P(B|A) .c. Find P(A ∩ B).d. Find P(Ac ∩ B).e. Find P(B) .f. Find P(A|B).g. Are Aand B independent? Is it reasonable to treat A and B as though they were independent? Explain.

Answer 1

Answer:

Step-by-step explanation:

a.

Remember that in general, the probability of an event is

(number of cases)/ (total number of cases)

Therefore for this part it is 350/1000 = 0.35

b.

If A occurs then that means that a 349 elements are defective out of 999, therefore

349/999 = 0.34

c.

Remember that

$P(A \cap B) = P(B|A) P(A) = (\frac{349}{999})*(\frac{35}{1000})$

(349/999)*(35/1000) = 0.01

d.

You are computing the probability that your first component is not defective and the second is, the probability that the first is not defective is 650/1000 and the probability that the second is defective is 350/999 therefore for the answer is

(650/1000)*(450/999) = 0.29

e.

Remember that

$P(B) = P(B \cap A)+P(B \cap A^c) = 0.01+0.29 = 0.3$

f.

For this part your can use Bayes theorem

 ${\displaystyle P(A|B) = \frac{P(B|A)P(A)}{P(B)} = (0.34*0.35)/0.3 }$

= 0.39

g.

They are not at all. If one of them is defective that will definitely affect the probability of the other event.