Consider two events a and b you are told that pa 051 that pb...

Consider two events, A and B. You are told that P(A) = 0.51, that P(B) = 0.87, and that P(B ∩ A') ≥ 0.02. (a). What is the maximum possible value for P(A ∩ B)? Justify. (b). What is the minimum possible value for P(A ∩ B)? Justify. (c). What is the maximum possible value for P(A ∪ B)? Justify. (d). What is the minimum possible value for P(A ∪ B)? Justify. (e). What is the maximum possible value for P(A ∪ B) - P(A ∩ B)? Hint: Use a Venn diagram to help you. Here, "possible value" is a value that can actually be achieved (it is possible to equal the value – e.g., we would not say that the "maximum possible value" of P(A ∪ B) is 1.2, since, while it is true that P(A ∪ B) ≤ 1.2, it is never possible for P(A ∪ B) to equal 1.2, under any circumstances).

Transcript

00:01 So this question we have that the probability of a is 0 .51, the probability of b is 0 .87, and the probability of b and not a is greater than or equal to 0 .0 .02.

00:20 So first of all, what's the maximum possible value for a intersection b? well, the probability of a intersection, so the probability of a union b is the probability of a plus the probability of b minus the probability of a intersection b.

00:49 And this is going to put a bound on the probability of a into section b because the probability of a and b has got to be less than or equal to 1.

01:00 So 1 is greater than or equal to the probability of a plus the probability of b minus the probability of a intersection b.

01:08 So the probability of a intersection b has got to be greater than or equal to the probability of a plus the probability of b minus 1.

01:17 So it's greater than or equal to 0 .51 plus 0 .87 minus 1, which is 0 .38.

01:26 It's the probability of a intersection b.

01:34 So that puts a minimum value on it.

01:37 So this is actually the answer for part b.

01:41 But now we want to put a maximum value on it as well.

01:44 Well, the probability of b is the probability of a intersection b plus the probability of not a intersection b.

01:57 So then the probability of a intersection b is the probability of b minus the probability of b intersection not a.

02:06 But b intersection not a is greater than or equal to 0 .0 .02.

02:12 So this is less than or equal to the probability of b minus 0 .02.

02:18 So we have the probability of a intersection b is less than or equal to 0 .87 minus 0 .02, which is 0 .85.

02:31 Okay.

02:32 So that is the answer for part a.

02:36 So then between a and b, we have that 0 .38 less than or equal to the probability of a intersection b, less than are equal to 0 .85...

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