Events A and B are mutually exclusive. Suppose event A occurs with probability 0.05 and event B occurs with probability 0.87.Compute the following.(If necessary, consult a list of formulas.)(a) Compute the probability that A does not occur or B does not occur (or both).(b) Compute the probability that either A occurs without B occurring or A and B both occur.
Question
Events A and B are mutually exclusive. Suppose event A occurs with probability 0.05 and event B occurs with probability 0.87.Compute the following.(If necessary, consult a list of formulas.)(a) Compute the probability that A does not occur or B does not occur (or both).(b) Compute the probability that either A occurs without B occurring or A and B both occur.
Solution
(a) To compute the probability that A does not occur or B does not occur (or both), we can use the formula for the union of two events. The formula is P(A ∪ B) = P(A) + P(B) - P(A ∩ B), where P(A) represents the probability of event A occurring, P(B) represents the probability of event B occurring, and P(A ∩ B) represents the probability of both events A and B occurring simultaneously.
Given that events A and B are mutually exclusive, meaning they cannot occur at the same time, we know that P(A ∩ B) = 0. Therefore, the formula simplifies to P(A ∪ B) = P(A) + P(B).
Substituting the given probabilities, we have P(A ∪ B) = 0.05 + 0.87 = 0.92.
Therefore, the probability that A does not occur or B does not occur (or both) is 0.92.
(b) To compute the probability that either A occurs without B occurring or A and B both occur, we can use the formula for the union of two events. The formula is P(A ∪ B) = P(A) + P(B) - P(A ∩ B), where P(A) represents the probability of event A occurring, P(B) represents the probability of event B occurring, and P(A ∩ B) represents the probability of both events A and B occurring simultaneously.
Given that events A and B are mutually exclusive, meaning they cannot occur at the same time, we know that P(A ∩ B) = 0. Therefore, the formula simplifies to P(A ∪ B) = P(A) + P(B).
Substituting the given probabilities, we have P(A ∪ B) = 0.05 + 0.87 = 0.92.
Therefore, the probability that either A occurs without B occurring or A and B both occur is 0.92.
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