Let P(A) = 0.35, P(B) = 0.30, and P(A ∩ B) = 0.17.
a. Are A and B independent events?
multiple choice 1
Yes because P(A | B) = P(A).
Yes because P(A ∩ B) ≠ 0.
No because P(A | B) ≠ P(A).
No because P(A ∩ B) ≠ 0.
b. Are A and B mutually exclusive events?
multiple choice 2
Yes because P(A | B) = P(A).
Yes because P(A ∩ B) ≠ 0.
No because P(A | B) ≠ P(A).
No because P(A ∩ B) ≠ 0.
c. What is the probability that neither A nor B takes place? (Round your answer to 2 decimal places.)
c. The probability that neither A nor B takes place can be found using the formula:
P(A' ∩ B') = 1 - P(A ∪ B)
We can find P(A ∪ B) using the formula:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Given that P(A) = 0.35, P(B) = 0.30, and P(A ∩ B) = 0.17, we can substitute these values into the formula:
P(A ∪ B) = 0.35 + 0.30 - 0.17 = 0.48
Now we can find P(A' ∩ B'):
P(A' ∩ B') = 1 - P(A ∪ B) = 1 - 0.48 = 0.52
Therefore, the probability that neither A nor B takes place is 0.52.
a. No because P(A | B) ≠ P(A).
b. No because P(A ∩ B) ≠ 0.
c. To find the probability that neither A nor B takes place, we can use the formula:
P(A' ∩ B') = P(S) - P(A ∪ B) (where S is the sample space and ∪ represents union)
Since events A and B are not mutually exclusive, we have:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Substituting the given values:
P(A ∪ B) = 0.35 + 0.30 - 0.17 = 0.48
Therefore,
P(A' ∩ B') = 1 - P(A ∪ B) = 1 - 0.48 = 0.52
The probability that neither A nor B takes place is 0.52.
To determine whether events A and B are independent, we need to compare the probability of A given B (P(A | B)) with the probability of A (P(A)). If P(A | B) = P(A), then the events are independent.
To calculate P(A | B), we can use the formula P(A | B) = P(A ∩ B) / P(B) since P(A ∩ B) = 0.17 and P(B) = 0.30.
P(A | B) = 0.17 / 0.30 = 0.57
Next, we compare P(A | B) with P(A). If P(A | B) = P(A), then events A and B are independent.
P(A) = 0.35
Since P(A | B) does not equal P(A) (0.57 ≠ 0.35), events A and B are not independent. Therefore, the correct answer for part a is "No because P(A | B) ≠ P(A)."
To determine whether events A and B are mutually exclusive, we need to check if their intersection (A ∩ B) is equal to zero.
Since P(A ∩ B) = 0.17, which is not equal to zero, events A and B are not mutually exclusive. Therefore, the correct answer for part b is "No because P(A ∩ B) ≠ 0."
Now, to find the probability that neither A nor B takes place, we can use the complement rule. The complement of event A is denoted as A' (not A), and the complement of event B is denoted as B' (not B).
P(A') = 1 - P(A) = 1 - 0.35 = 0.65
P(B') = 1 - P(B) = 1 - 0.30 = 0.70
Since A and B are not mutually exclusive, we can use the formula P(A' ∩ B') = 1 - P(A ∪ B) to find the probability that neither A nor B takes place.
P(A' ∩ B') = 1 - (P(A) + P(B) - P(A ∩ B))
P(A' ∩ B') = 1 - (0.35 + 0.30 - 0.17)
P(A' ∩ B') = 1 - 0.48
P(A' ∩ B') = 0.52
Therefore, the probability that neither A nor B takes place is 0.52.