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Numerade Educator



Problem 14 Medium Difficulty

Consider randomly selecting a student at a certain university, and let $A$ denote the event that the
selected individual has a Visa credit card and $B$ be the analogous event for a MasterCard. Suppose that $P(A)=.5, P(B)=.4,$ and $P(A \cap B)=.25 .$
(a) Compute the probability that the selected individual has at least one of the two types of cards (i.e., the probability of the event $A \cup B )$ .
(b) What is the probability that the selected individual has neither type of card?
(c) Describe, in terms of $A$ and $B$ , the event that the selected student has a Visa card but not a MasterCard, and then calculate the probability of this event.


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Video Transcript

any question? Fortin gonna look at students at university picked randomly and see whether or not the album he's, um, or a MasterCard or the boat. Um, so they're given they give us a couple probabilities to go along. This so probability that someone has a visa right Like that. That probability is 0.5. The probability that a student has a MasterCard is point for and then the probability that they have both Hey, can be that is evil. The 0.25 All right, So, hey, compute the probability probability that the selected individual has at least one of the two types of cards. AII the probability event a union Be right. So let's try a little Ben diagram here. Toe resent What's going on? So here, event A. So the people in this one have seasons. These people have massacres. And the part and teach me Didn't hear they have. Okay. Now, um, the probability that you are in this circle over here is went by. Probably you're in this surf of year is 0.4, and the probability you're in both circles this 0.25 now, they want us to find the probability that you're either one of the two circles. Hey, so find the probability of a union. So what's the probability of that? You're in any of these locations here? All right, So, basically, if you were to just say, the probability of a was probably be what? You'd be pretty darn close. But here's the idea. If the probability is this area here and the probability of B is this area year, you see how we counted this part right here twice? Okay, that basically, when you add that, probably a at the problem together, you're double counting the intersection. So we actually have to subtract from that. The intersection part of day, which is probably that is 0.25 All right, so we end up getting 0.9 minus 0.25 So the probability of a union be his 0.65 All right, so then the next part says, What's the probability that selected individuals as either type of car? Okay, so that would be anyone who's outside of both these circles. So the probability of neither card neither, um would be one minus 100%. I mind this point spot. So that would be point, Reba Yeah, All right, then Finally see described in terms that you can be the event that selected student has a visa card, but not in MasterCard. And he calculated probability of this. So has a visa, but not a master. Right. So as a visa means you're in a but not abuse in submarines. We want to have this area year. Okay, so we don't wanna have this area year. He is not. All right, So, um, we want to know where, um, Just this part, See? All right, so steers the probability that you have a visa card, but not a MasterCard. So we wanna do is want intersect A with all the people who don't have be there will be a be there. Right? So basically, it's gonna be point, but we want to take out 0.25 That's, um um, that's the overlapping here. 25 that out. So our final answer would be to five