Assume an office has two telephones (A and B) with separate lines and different telephone numbers. The probability that telephone A rings when the office is closed is .30. The probability that telephone B rings when the office is closed is .20.
What is the probability that telephone A or B rings when the office is closed?
Based on what I did for the other two can you figure out this one?
What is your answer?
Either-or probabilities are found by adding the individual probabilities.
To find the probability that either telephone A or B rings when the office is closed, we will use the concept of union in probability. The union of two events A and B, denoted as A ∪ B, represents the event that either A or B occurs (or both). In this case, the "ringing of telephone A when the office is closed" and the "ringing of telephone B when the office is closed" are the two events.
To calculate the probability of the union of two events, we use the formula:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Where P(A) represents the probability of event A, P(B) represents the probability of event B, and P(A ∩ B) represents the probability of the intersection of events A and B (meaning both A and B occurring simultaneously).
In this case, we are given the probability of telephone A (P(A)) ringing when the office is closed as 0.30, and the probability of telephone B (P(B)) ringing when the office is closed as 0.20. However, we do not have information about the probability of both telephones ringing together (P(A ∩ B)).
Since no information is given about the relationship between the two events, we assume they are independent. Therefore, the ringing of telephone A and telephone B are independent events. In such cases, the probability of their intersection (P(A ∩ B)) is the product of their individual probabilities.
Hence, assuming independence between the two events:
P(A ∩ B) = P(A) * P(B)
Substituting the given probabilities:
P(A ∩ B) = 0.30 * 0.20
P(A ∩ B) = 0.06
Now, we can substitute this value back into the original formula:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A ∪ B) = 0.30 + 0.20 - 0.06
P(A ∪ B) = 0.44
Therefore, the probability that either telephone A or B rings when the office is closed is 0.44 or 44%.