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 find the probabilities in this scenario, you can use set theory and the concept of conditional probability. Let's break down each question and explain how to approach it.

a. If a product failed check A, what is the probability that it also failed check B?

To find the probability that a product failed both checks A and B, we need to use the concept of conditional probability. The formula for conditional probability is:

P(A and B) = P(A) * P(B|A)

In this case, P(A and B) represents the probability that a product failed both checks A and B, P(A) represents the probability that a product failed check A, and P(B|A) represents the probability that a product failed check B given that it failed check A.

From the information provided, we know that 3% failed both checks A and B, and 10% failed check A. Therefore:

P(A and B) = 3% (or 0.03)
P(A) = 10% (or 0.10)

Now, we can use the formula to find P(B|A):

P(A and B) = P(A) * P(B|A)
0.03 = 0.10 * P(B|A)

Rearranging the equation, we can solve for P(B|A):

P(B|A) = 0.03 / 0.10
P(B|A) = 0.3 (or 30%)

Therefore, the probability that a product failed check B given that it failed check A is 30%.

b. If a product failed check B, what is the probability that it also failed check A?

This question is similar to the previous one, but the roles of checks A and B are reversed. To find the probability that a product failed check A given that it failed check B, we again use conditional probability:

P(B and A) = P(B) * P(A|B)

In this case, P(B and A) represents the probability that a product failed both checks A and B, P(B) represents the probability that a product failed check B, and P(A|B) represents the probability that a product failed check A given that it failed check B.

From the information provided, we know that 3% failed both checks A and B, and 12% failed check B. Therefore:

P(B and A) = 3% (or 0.03)
P(B) = 12% (or 0.12)

Now, we can use the formula to find P(A|B):

P(B and A) = P(B) * P(A|B)
0.03 = 0.12 * P(A|B)

Rearranging the equation, we can solve for P(A|B):

P(A|B) = 0.03 / 0.12
P(A|B) = 0.25 (or 25%)

Therefore, the probability that a product failed check A given that it failed check B is 25%.

c. What is the probability that a product failed either check A or check B?

To find the probability that a product failed either check A or check B, we can use the principle of inclusion-exclusion:

P(A or B) = P(A) + P(B) - P(A and B)

From the information provided, we know that 10% failed check A, 12% failed check B, and 3% failed both checks A and B. Therefore:

P(A) = 10% (or 0.10)
P(B) = 12% (or 0.12)
P(A and B) = 3% (or 0.03)

Now, we can use the formula to find P(A or B):

P(A or B) = P(A) + P(B) - P(A and B)
P(A or B) = 0.10 + 0.12 - 0.03
P(A or B) = 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 the probability that a product failed neither check A nor check B, we subtract the probability of failure from the total:

P(neither A nor B) = 1 - P(A or B)

From the previous step, we found that P(A or B) = 0.19. Therefore:

P(neither A nor B) = 1 - P(A or B)
P(neither A nor B) = 1 - 0.19
P(neither A nor B) = 0.81 (or 81%)

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