17. Let A denote the event that the next request for assis- tance from a statistical software consultant relates to the SPSS package, and let B be the event that the next request is for help with SAS. Suppose that P(A) = .30 and P(B) = .50. a. Why is it not the case that P(A) + P(B) = 1? b. Calculate P(A'). C. Calculate P(A U B). d. Calculate P(A' B').
Could you double-check if my answers are correct for this question 2.17? If they are not correct, please point out the error and present the correction.
Question. 2.17:
a. P(A) + P(B) does not necessarily equal 1 because the events A and B could potentially overlap. In other words, it's possible for a request to be related to both the SPSS package and SAS simultaneously. The formula P(A) + P(B) - P(A ∩ B) gives the correct probability of either
b. The complement of event A, denoted as A', is the event that the next request is not related to the SPSS package. The probability of the complement of A can be calculated as:
P(A') = 1 - P(A)
= 1 - 0.30
= 0.70
c. The union of events A and B, denoted as A ∪ B, is the event that the next request is related to either the SPSS package or SAS or both. The probability of the union of A and B can be calculated using the formula:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
= 0.30 + 0.50 - P(A ∩ B)
However, I don't have information about the probability of the intersection of A and B (P(A ∩ B)), so I can't compute the exact value of P(A ∪ B) with the given information.
d. The complement of both events A and B, denoted as A' ∩ B', is the event that the next request is not related to the SPSS package (A') and is also not related to SAS (B'). The probability of the intersection of the complements of A and B can be calculated as:
P(A' ∩ B') = P(A' ∩ B)
= P(B) - P(A ∩ B)
= 0.50 - P(A ∩ B)
From the above given data the following solution is provided below:
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