At a quality control checkpoint on a manufacturing assembly line, 8% of the items failed check A, 10% failed check B, and 2% failed both checks A and B.

a. If a product failed check A, what is the probability that it also failed check B?
b. If a product failed check B, what is the probability that it also failed check A?
c. What is the probability that a product failed either check A or check B?
d. What is the probability that a product failed neither check A nor check B?

Failed A only: 6&

Failed B only: 8%
Failed A and B: 2%
Failied neither: 84%

a. 2/8 = 1/4 = 25%
b. 2/10 = 1/5 = 20%
c. 6% + 8% + 2% = 16%
d. 100% - 16% = 84%

To solve this problem, we can use conditional probability and set operations. Let's denote the events as follows:

A: The product failed check A.
B: The product failed check B.

a. To find the probability that a product failed check B given that it failed check A, we need to find P(B|A).

We know that P(A) = 0.08 (8% failed check A) and P(A ∩ B) = 0.02 (2% failed both checks A and B).

The formula for conditional probability is: P(B|A) = P(A ∩ B) / P(A).

So, the probability that a product failed check B given that it failed check A is: P(B|A) = P(A ∩ B) / P(A) = 0.02 / 0.08 = 0.25 or 25%.

b. To find the probability that a product failed check A given that it failed check B, we need to find P(A|B).

Similarly, P(B) = 0.10 (10% failed check B).

The formula for conditional probability is: P(A|B) = P(A ∩ B) / P(B).

So, the probability that a product failed check A given that it failed check B is: P(A|B) = P(A ∩ B) / P(B) = 0.02 / 0.10 = 0.2 or 20%.

c. To find the probability that a product failed either check A or check B, we need to calculate P(A U B), which represents the union of events A and B.

We know that P(A) = 0.08 and P(B) = 0.10.

The formula for the union of two events is: P(A U B) = P(A) + P(B) - P(A ∩ B).

So, the probability that a product failed either check A or check B is: P(A U B) = P(A) + P(B) - P(A ∩ B) = 0.08 + 0.10 - 0.02 = 0.16 or 16%.

d. To find the probability that a product failed neither check A nor check B, we need to calculate the complement of the event A U B.

The complement of an event A is denoted as Ā and represents the event "A does not occur."

The formula for the complement of an event is: P(Ā) = 1 - P(A U B).

So, the probability that a product failed neither check A nor check B is: P(Ā) = 1 - P(A U B) = 1 - 0.16 = 0.84 or 84%.

Therefore, the answers are:
a. The probability that a product failed check B given that it failed check A is 0.25 or 25%.
b. The probability that a product failed check A given that it failed check B is 0.2 or 20%.
c. The probability that a product failed either check A or check B is 0.16 or 16%.
d. The probability that a product failed neither check A nor check B is 0.84 or 84%.