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?

I am not good with probability and the percentages are really throwing me off. Any help would be great.

To solve this problem, we can use the principles of probability and set notation. Let's break down the problem 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 out of the total products that failed check A, we divide the number of products that failed both checks A and B by the total number of products that failed check A.

Let's assume there are 1000 products in total. If 8% failed check A, that would be 0.08 * 1000 = 80 products. And if 2% failed both checks A and B, that would be 0.02 * 1000 = 20 products.

Therefore, the probability that a product failed both checks A and B given that it failed check A is 20/80 = 0.25 or 25%.

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

We can use the same approach as in part a, but focusing on the probability of failing check A given that check B has failed.

Using the given percentages, 10% of the total products failed check B, which would be 0.1 * 1000 = 100 products. And since 2% of the products failed both checks A and B, that would be 0.02 * 1000 = 20 products.

Therefore, the probability that a product failed both checks A and B given that it failed check B is 20/100 = 0.2 or 20%.

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 add the probabilities of failing check A and failing check B, and then subtract the probability of failing both checks A and B since it was counted twice.

Using the given percentages, 8% of the total products failed check A (80 products) and 10% of the total products failed check B (100 products). However, since 2% of the products failed both checks A and B, we need to subtract that to avoid double-counting.

Therefore, the probability that a product failed either check A or check B is (8 + 10 - 2)% = 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 can subtract the probability of failing either check A or check B from 100%.

Using the given percentages, the probability of failing either check A or check B is 16%. Therefore, the probability of not failing either check A or check B is 100% - 16% = 84%.

Therefore, the probability that a product failed neither check A nor check B is 84%.

To solve these probability questions, we can use the concept of conditional probability, the addition rule, and the complement rule. Let's break down each part of the question.

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 check B given that it failed check A, we need to use conditional probability. We know that 2% of the products failed both checks A and B, and 8% of the products failed check A. Therefore, the number of products that failed check A but also failed check B is 2% of 8%. So, the probability is:

P(B | A) = (probability of failing both checks) / (probability of failing check A)
= 2% / 8%
= 0.02 / 0.08
= 0.25

So, there is a 25% probability that a product that failed check A also failed check B.

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 check A given that it failed check B, we use the same approach as in part a. We know that 2% of the products failed both checks A and B, and 10% of the products failed check B. Therefore, the number of products that failed check B but also failed check A is 2% of 10%. So, the probability is:

P(A | B) = (probability of failing both checks) / (probability of failing check B)
= 2% / 10%
= 0.02 / 0.1
= 0.2

So, there is a 20% probability that a product that failed check B also failed check A.

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 addition rule. We know that 8% of the products failed check A, 10% failed check B, and 2% failed both checks. To calculate the probability of failing either check A or check B, we add the probabilities of failing each individual check and subtract the probability of failing both checks to avoid double counting:

P(A or B) = P(A) + P(B) - P(A and B)
= 8% + 10% - 2%
= 0.08 + 0.1 - 0.02
= 0.16

So, there is a 16% 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 probability that a product failed neither check A nor check B, we can use the complement rule. The complement rule states that the probability of an event not occurring is 1 minus the probability of the event occurring. In this case, the event we are interested in is the product failing either check A or check B. So, the probability of the product passing both checks is:

P(neither A nor B) = 1 - P(A or B)
= 1 - 0.16
= 0.84

So, there is an 84% probability that a product failed neither check A nor check B.