Let A and B be independent events with P(A) = 0.40 and P(B) = 0.50.
a. Calculate P(A ∩ B). (Round your answer to 2 decimal places.)
b. Calculate P((A U B)c). (Round your answer to 2 decimal places.)
c. Calculate P(A | B). (Round your answer to 2 decimal places.)
a. P(A ∩ B) = P(A) * P(B) = 0.40 * 0.50= 0.20
b. P((A U B)c) = 1 - P(A U B)
Since A and B are independent, P(A U B) = P(A) + P(B) - P(A ∩ B)
= 0.40 + 0.50 - 0.20 = 0.70
Therefore, P((A U B)c) = 1 - 0.70 = 0.30
c. P(A | B) = P(A ∩ B) / P(B)
= 0.20 / 0.50 = 0.40
To solve these questions, we will use the following formulas:
a. P(A ∩ B) = P(A) * P(B)
b. P((A U B)c) = 1 - P(A U B)
c. P(A | B) = P(A ∩ B) / P(B)
Now let's calculate each part step by step:
a. P(A ∩ B) = P(A) * P(B)
= 0.40 * 0.50
= 0.20
b. P((A U B)c) = 1 - P(A U B)
However, to calculate P(A U B), we need to know if A and B are mutually exclusive or not.
If A and B are mutually exclusive, then P(A U B) = P(A) + P(B).
If A and B are not mutually exclusive, then P(A U B) = P(A) + P(B) - P(A ∩ B).
Since we don't have any information about the mutual exclusivity of A and B, we'll assume they are not mutually exclusive. Thus,
P((A U B)c) = 1 - P(A U B)
= 1 - (P(A) + P(B) - P(A ∩ B))
= 1 - (0.40 + 0.50 - 0.20)
≈ 0.30
c. P(A | B) = P(A ∩ B) / P(B)
= 0.20 / 0.50
≈ 0.40
To solve these questions, we need to use basic probability formulas and the definition of independence.
a. To calculate P(A ∩ B), we use the formula:
P(A ∩ B) = P(A) * P(B)
Since A and B are independent events, we can multiply their probabilities:
P(A ∩ B) = 0.40 * 0.50 = 0.20
So, P(A ∩ B) is 0.20.
b. To calculate P((A U B)c), we can use the complement rule, which states that the probability of the complement of an event is equal to 1 minus the probability of the event itself:
P((A U B)c) = 1 - P(A U B)
Since A and B are independent, we can use the addition rule:
P((A U B)c) = 1 - P(A) - P(B)
Note that P(A U B) is the probability of either A or B or both occurring. Since A and B are independent, the probability of either A or B or both occurring can be calculated as:
P(A U B) = P(A) + P(B) - P(A ∩ B)
Using the values provided:
P((A U B)c) = 1 - 0.40 - 0.50 + 0.20 = 0.30
So, P((A U B)c) is 0.30.
c. To calculate P(A | B), we use the conditional probability formula:
P(A | B) = P(A ∩ B) / P(B)
Using the value of P(A ∩ B) calculated in part a and the provided value of P(B):
P(A | B) = 0.20 / 0.50 = 0.40
So, P(A | B) is 0.40.