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 solve these questions, we can use the concept of conditional probability, which measures the probability of an event occurring given that another event has already occurred.

Let's denote event A as "product failed check A" and event B as "product failed check B."

a. To find the probability that a product failed check B given that it failed check A, we can use the formula for conditional probability:

P(B|A) = P(A∩B) / P(A)

Here, P(A∩B) represents the probability that a product failed both check A and check B, and P(A) represents the probability that a product failed check A.

Given that 3% of the products failed both checks A and B, and 10% of the products failed check A, we have:

P(B|A) = (3% of total products) / (10% of total products)

b. Similarly, to find the probability that a product failed check A given that it failed check B, we can use the same formula:

P(A|B) = P(A∩B) / P(B)

Here, P(A∩B) represents the probability that a product failed both checks A and B, and P(B) represents the probability that a product failed check B.

Given that 3% of the products failed both checks A and B, and 12% of the products failed check B, we have:

P(A|B) = (3% of total products) / (12% of total products)

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

P(A∪B) = P(A) + P(B) - P(A∩B)

Here, P(A) represents the probability that a product failed check A, P(B) represents the probability that a product failed check B, and P(A∩B) represents the probability that a product failed both checks A and B.

Given that 10% of the products failed check A, 12% of the products failed check B, and 3% of the products failed both checks A and B, we have:

P(A∪B) = (10% of total products) + (12% of total products) - (3% of total products)

d. Lastly, to find the probability that a product failed neither check A nor check B, we can use the complement rule:

P(~A∩~B) = 1 - P(A∪B)

Here, P(~A∩~B) represents the probability that a product passed both checks A and B (i.e., it didn't fail either check). Hence, the probability that a product failed neither check A nor check B is the complement of that.

Using the result from part (c), we have:

P(~A∩~B) = 1 - P(A∪B)

To solve these questions, we need to use the concept of conditional probability. Let's go through each question step-by-step:

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 given that it failed check A, we need to divide the number of products that failed both checks by the number of products that failed check A.

Let's assume we have 1000 products in total.
The number of products that failed check A is 10% of 1000 = 0.1 * 1000 = 100.
The number of products that failed both checks A and B is 3% of 1000 = 0.03 * 1000 = 30.

So, the probability that a product failed both checks A and B given that it failed check A is 30/100 = 0.3, or 30%.

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

To find the probability that a product failed both checks A and B given that it failed check B, we need to divide the number of products that failed both checks by the number of products that failed check B.

Using the same assumption as before, 1000 products in total:
The number of products that failed check B is 12% of 1000 = 0.12 * 1000 = 120.
The number of products that failed both checks A and B is still 30.

So, the probability that a product failed both checks A and B given that it failed check B is 30/120 = 0.25, or 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 need to add the probabilities of the two events and subtract the probability that the product failed both checks.

Using the same assumptions as before:
The probability that a product failed check A is 10% = 0.1.
The probability that a product failed check B (excluding the ones that also failed check A) is 12% - 3% = 9% = 0.09.
The probability that a product failed both checks A and B is 3% = 0.03.

The probability that a product failed either check A or check B is 0.1 + 0.09 - 0.03 = 0.16, or 16%.

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 need to subtract the probability that it failed either check A or check B from 1 (since the sum of all probabilities should equal 1).

Using the same assumptions as before:
The probability that a product failed either check A or check B is 0.16.

The probability that a product failed neither check A nor check B is 1 - 0.16 = 0.84, or 84%.