In the manufacturing of a chemical adhesive, 9% of all batches have raw materials from two different lots. This occurs when holding tanks are replenished and the remaining portion of a lot is insufficient to fill the tanks. Only 5% of batches with material from a single lot require reprocessing. However, the viscosity of batches consisting of two or more lots of material is more difficult to control, and 40% of such batches require additional processing to achieve the required viscosity. Let A denote the event that a batch is formed from two different lots, and let B denote the event that a lot requires additional processing. Determine the following probabilities.
a. P(A)
b. P(A')
c. P(B\A)
d. P(B\A')
e. P(A ∩ B)
f. P(A ∩ B')
g. P(B)

Step-by-step explanation:

Hello!

In a chemical adhesive manufacturing company you can encounter the following situations:

The adhesive is made in batches.

9% of all batches have raw materials from two different lots.

5% of all batches with material from a single lot require reprocessing.

40% of all batches with the material of two or more lots require additional processing.

You have two events:

A: A batch is formed with two different lots.

B: The lot requires additional processing.

And you can define their complements:

A': A batch is formed with one lot.

B': The lot does not require additional processing.

a. This probability is given in the text "9% of all batches have raw materials from two different lots. ", symbolically:

P(A)= 0.09

b. If the probability of a batch containing two lots is 9%, then it's complimentary event has a probability equal to:

P(A')= 1 - 0.09= 0.91

c. P(B/A) It is a conditional probability and you can interpret it as "the lot requires additional processing given that the batch is formed with two lots"

This information is also given:

P(B/A)= 0.40

d. P(B/A') Is the probability of the lot requiring additional processing given that the batch is formed with one lot.

P(B/A')= 0.05

e. Now you have to calculate the probability that "the batch is made with two lots" and "the lot requires additional processing"

A and B aren't independent events, which means that the occurrence of one of them modifies the probability of occurrence of the other one in two replications of the experiment.

P(A ∩ B)= P(A) * P(B/A)

P(A ∩ B)= 0.09 * 0.40= 0.036

f. You have to calculate the probability of "batch is formed with two different lots" and "the lot does not require additional processing", these two events aren't independent.

You cannot calculate the probability of this intersection the same way you did in item e. because we have no information about the probability of P(B'/A). But the P(A) is equal to the summation of P(A ∩ B) and P(A ∩ B'), so:

P(A ∩ B')= P(A) - P(A ∩ B)= 0.09 - 0.036= 0.054

g. To calculate the probability of the event "The lot requires additional processing." you have to add P(A ∩ B)+P(A' ∩ B)

First calculate P(A' ∩ B) = P(A')* P(B/A')= 0.91*0.05= 0.0455

P(B)= P(A ∩ B) + P(A' ∩ B)= 0.036 + 0.0455= 0.0815

I hope you have a SUPER day!