A fair six sided die is rolled describe the sample space s...

A fair, six-sided die is rolled. Describe the sample space S, identify each of the following events with a subset of S, and compute its probability (an outcome is the number of dots that show up). S = {1, 2, 3, 4, 5, 6} A. A = {2, 4, 6} P(A) = B. B = P(B) = C. A' = P(A') = D. P(A | B) = E. P(B | A) = F. A ∩ B = P(A ∩ B) = G. P(A ∩ B) = H. A ∪ B = P(A ∪ B) = I. P(A ∪ B') = A = {2, 4, 6} B = {1, 2, 3} B' = {4, 5, 6}

Transcript

00:01 We're looking at the rolling of a fair six -sided die.

00:05 So we have the sample space, which is 1, 2, 3, 4, 5 and 6, and we want to work out the probability of a variety of events.

00:15 Okay, so start with a.

00:17 So a is 2, 4 or 6.

00:20 So that's going to be a 1 out of 2 chance, because there are 6 possible outcomes.

00:26 Each outcome has a 1 in 6 chance.

00:28 So we'll put that.

00:29 Each outcome is 1 in 6.

00:35 So what we have to really do here is find out how many different outcomes an event includes.

00:41 A includes three different outcomes, so that's 3 out of 6.

00:46 B is 1, 2 or 3.

00:49 That's also going to be 1 in 2.

00:53 C, looking at the complement to a.

00:57 Okay.

00:58 So the complement to a -dash is anything that does not happen in a.

01:04 So where a is 2, 4 and 6, a with the complement is going to be 1, 3 and 5, which also has a 1 and 2 chance of happening.

01:20 Notice that a and the complement to a probabilities add up to 1 because between them they include the entire sample space.

01:28 For d, we want the probability of a given b.

01:34 So i'm going to write out another formula here.

01:38 I'm going to use x and y.

01:40 X given y is equal to the intersect of them divided by the probability of y.

01:51 So here, the intersect, the things that occur in both of them, both a and b, would be what? so a is 2 ,4 and 6, b is 1, 2 and 3, so it's just 2.

02:05 So that would be 1 and 6 intersect over 1 and 2, which is the probability of b.

02:13 So that would be, oops, 1 in 12.

02:20 It seems i don't know, no, no, no, 1 in 3.

02:24 There we go.

02:29 Now we look at part e, and we want probability of b given a.

02:38 So we already have the intersect, and a is a half, so this is also going to be a third...