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?
a. 10% out of 100% failed A or 10/100
3% of 100% failed A & B or 3/100
3% that failed A(10%) also failed B or 3/10
b. 12% out of 100% failed B
3% out of 100% failed A & B or 3/12
c. failed A - 10%
failed B - 12%
sub total 22%(-3% counted double-failed both)
= 19% total
d. 100%-19% that failed =81% chance product did not fail.
To solve these probability questions, we can use the concept of conditional probability. Let's denote the events as follows:
A = The product failed check A
B = The product failed check B
Now let's solve 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 check B given that it failed check A, we need to use conditional probability. We can use the formula:
P(B|A) = P(A and B) / P(A)
We are given that 3% failed both checks A and B, so P(A and B) = 0.03.
We also know that 10% failed check A, so P(A) = 0.10.
Therefore, the probability that a product failed check B given that it failed check A is:
P(B|A) = 0.03 / 0.10 = 0.3 or 30%.
b. If a product failed check B, what is the probability that it also failed check A?
Using the same conditional probability formula:
P(A|B) = P(A and B) / P(B)
We already have P(A and B) = 0.03.
And we are given that 12% failed check B, so P(B) = 0.12.
Thus, the probability that a product failed check A given that it failed check B is:
P(A|B) = 0.03 / 0.12 = 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 can use the principle of inclusion-exclusion:
P(A or B) = P(A) + P(B) - P(A and B)
We know that 10% failed check A and 12% failed check B.
And 3% failed both checks A and B.
So, the probability that a product failed either check A or check B is:
P(A or B) = 0.10 + 0.12 - 0.03 = 0.19 or 19%.
d. What is the probability that a product failed neither check A nor check B?
The probability that a product failed neither check A nor check B is the complement of the probability that it failed either check A or check B:
P(neither A nor B) = 1 - P(A or B)
We already found that P(A or B) = 0.19.
Thus, the probability that a product failed neither check A nor check B is:
P(neither A nor B) = 1 - 0.19 = 0.81 or 81%.