Can someone please help me.

At a quality control checkpoint on a manafacturing assembly line, 10% of the items failed check A, 12% failed check B, and 3% 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?

To answer these questions, we'll need to use some concepts from probability theory.

Let's define the following events:
- Event A: Product failed check A
- Event B: Product failed check B

We are provided with the following information:
- P(A) = 10% = 0.10
- P(B) = 12% = 0.12
- P(A and B) = 3% = 0.03

Now, let's proceed with each question:

a. If a product failed check A, what is the probability that it also failed check B?
To find this probability, we need to use conditional probability. The probability of A and B happening together (P(A and B)) divided by the probability of A occurring (P(A)) will give us the answer.
P(A and B) / P(A) = 0.03 / 0.10 = 0.30 or 30%

Therefore, if a product failed check A, there is a 30% probability it also failed check B.

b. If a product failed check B, what is the probability that it also failed check A?
Similarly, we can use conditional probability to find this probability.
P(A and B) / P(B) = 0.03 / 0.12 = 0.25 or 25%

Therefore, if a product failed check B, there is a 25% probability it also failed check A.

c. What is the probability that a product failed either check A or check B?
To find the probability of the union of two events, we can add their individual probabilities and subtract the probability of their intersection.
P(A or B) = P(A) + P(B) - P(A and B) = 0.10 + 0.12 - 0.03 = 0.19 or 19%

Therefore, the probability that a product failed either check A or check B is 19%.

d. What is the probability that a product failed neither check A nor check B?
To find this probability, we subtract the probability of failing either check A or check B from 1 (since the sum of all probabilities must equal 1).
P(neither A nor B) = 1 - P(A or B) = 1 - 0.19 = 0.81 or 81%

Therefore, the probability that a product failed neither check A nor check B is 81%.

Please note that these calculations assume the events are statistically independent.