Problem 6 in an experiment a b c and d are events with...

Problem 6. In an experiment, A, B, C, and D are events with probabilities P(A U B) = 5/8; P(A) = 3/8, P(C ∩ D) = 1/3, and P(C) = 1/2. Furthermore, A and B are disjoint, while C and D are independent. a) Find P(A ∩ B), P(B), P(A ∩ B^c), and P(A U B^c). b) Are A and B independent? c) Find P(D), P[C ∩ D^c), P(C^c ∩ D^c), and P(C|D). d) Find P(C U D) and P(C U D^c). e) Are C and D^c independent?

Transcript

00:01 So we're giving all this information about sets a and b, and we have the union is 5 .8s.

00:06 We have the probability of a is 3 8s, and we're told that a and b are disjoint, meaning that their intersection is 0 as a probability of 0.

00:15 So we can answer that question to begin with.

00:18 Now, let's also draw the venn diagram drawn for this setting, and if this is event a and this is event b, the union is 5x, that means the probability.

00:30 Of event b must be these two eighths or one fourth now we have the outside of the ven diagram if these two add up to five eights then this area out here must be three eights so now we can answer the remainder of the questions we want to find the intersection of a with b complement and b complement is everything that's outside of b and that is actually the set a now we want to find the union of a and b complement.

01:06 So a is obviously a and b complement is everything that is outside of b.

01:12 So that includes everything but what is inside of b.

01:16 So that is six -eighths or three -fours.

01:21 And are a and b independent? our answer is no because we don't have the probability of the intersection does not equal the probability of each of these multiplied together.

01:35 So we don't have independence.

01:38 Now, looking at set d, we have a different scenario here, and we have the intersection of these two sets is one -third.

01:46 We know that event c is one -half, and we know that c and d are independent.

01:52 So we know that that means that the probability of c times the probability of d has to equal the intersection.

02:01 So that means the probability of d must be one -half, excuse me, one -third times the reciprocal of two over one, and that gives us a value of two -thirds.

02:16 So this is the probability of d.

02:18 Now, we already have one -third in here.

02:21 That means this part outside of c has to be one -third.

02:26 The sum of these two has to be the probability.

02:29 Probability of d.

02:30 We also know the sum of one -third plus one plus whatever this component is here has to add up to one -half.

02:41 And so let's write them as six.

02:44 And so we can see that that needs to be two six, excuse me, one -six.

02:52 So we have two -six plus one -six...