Let A and B be mutually exclusive events with P(A) = 0.26 and P(B) = 0.36.
a. Calculate P(A ∩ B). (Round your answer to 2 decimal places.)
b. Calculate P(A U B). (Round your answer to 2 decimal places.)
c. Calculate P(A | B). (Round your answer to 2 decimal places.)
a. Since A and B are mutually exclusive events, the intersection of A and B is empty, so P(A ∩ B) = 0.
b. The union of A and B consists of all outcomes that are in A or in B or in both. Since A and B are mutually exclusive, P(A U B) = P(A) + P(B) = 0.26 + 0.36 = 0.62.
c. The probability of A given B, denoted as P(A | B), is the probability of event A occurring given that event B has already occurred. Since A and B are mutually exclusive, if event B has occurred, then event A cannot occur. Thus, P(A | B) = 0.
To calculate the probabilities, we'll use the fact that for mutually exclusive events, the probability of their intersection is zero.
a. P(A ∩ B) = 0
b. To calculate P(A U B), we can use the formula: P(A U B) = P(A) + P(B) - P(A ∩ B). Since P(A ∩ B) = 0, we have P(A U B) = P(A) + P(B) = 0.26 + 0.36 = 0.62.
c. To calculate P(A | B), we can use the formula: P(A | B) = P(A ∩ B) / P(B). Since P(A ∩ B) = 0, we have P(A | B) = 0 / 0.36 = 0.
To calculate the probabilities mentioned in the question, we can use the basic principles of probability. Let's break down each calculation step by step:
a. To calculate P(A ∩ B), we need to find the probability of both events A and B occurring simultaneously. However, since A and B are mutually exclusive events, they cannot occur at the same time. This means that the intersection of A and B is an empty set, resulting in a probability of 0.
b. To calculate P(A U B) (the union of A and B), we need to find the probability of either event A or event B occurring. Since the events are mutually exclusive, the probability of either A or B occurring is the sum of their individual probabilities. Therefore, P(A U B) = P(A) + P(B) = 0.26 + 0.36 = 0.62.
c. To calculate P(A | B) (the probability of event A given that event B has occurred), we can use the formula: P(A | B) = P(A ∩ B) / P(B). However, since A and B are mutually exclusive, P(A ∩ B) = 0 (as determined in part a). Thus, P(A | B) = 0 / P(B) = 0 / 0.36 = 0.
Therefore, the answers to the calculations are:
a. P(A ∩ B) = 0
b. P(A U B) = 0.62
c. P(A | B) = 0