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Suppose that vehicles taking a particular freeway exit can turn right $(R),$ turn left $(L),$ or go straight $(S) .$ Consider observing the direction for each of three successive vehicles.

(a) List all outcomes in the event $A$ that all three vehicles go in the same direction.

(b) List all outcomes in the event $B$ that all three vehicles take different directions.

(c) List all outcomes in the event $C$ that exactly two of the three vehicles turn right.

(d) List all outcomes in the event $D$ that exactly two vehicles go in the same direction.

(e) List the outcomes in $D^{\prime}, C \cup D,$ and $C \cap D .$

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all right. A problem for we have three vehicles taking a freeway exit at the end that neither turn right left or go straight. They want us to list all the outcomes in the event that all three vehicles go in the same direction. Okay, so those outcomes would be first car left. Second car left there are left, or we're a scar. Right? Set your car right there are right. Or lastly, first car, straight, second car. Destry your Card street. All right, I'm b list thea Outcomes in the event that be, uh, that they're all going different directions. Okay, so we need to do is mess around with the order of ours. L's and s is all right, so I'm gonna say the first part turns right. So that means the second car in turn, left in the 3rd 1 straight. Um, but I could also take these first last to reverse the order so you could have right for the first car in st with the second car. You left with 1/3. So that's all of the possibilities for when the first car goes right? First car goes right. It's not gonna do the same thing this time the first person left. So then the other two might go right straight, or the other two might go straight and right. You see, I'm just switching the order of the 2nd 2 So those are all the possibilities where the first cargo's left last is? Yes. 1/3 car goes straight up. Sorry. He left. It's the street with a right. Left, um, straight with left, right? Yeah. And we've got all six possibilities. Okay, um, see, list all outcomes in the event that see that exactly two of the three vehicles turn right. Two of the three turn right. Okay, so you started out with, um let me see. 1st 1 goes, right? Yeah. Then the 2nd 1 does, Uh, well oh, right. Less right. Yeah. You can follow that up with, huh? Right straight. Right. And then, Right, Right. Left, right, Right, Street. Now, the only place left to put the two to ours with this one right here would be to put them in the last two position so that the l r what street are. Okay, so that's the, um, uh, the probability that, um, two of the three vehicles turn right Okay, So now, Indy, they switch it up and they say What? Yeah, you missed all the outcomes where exactly two vehicles go in the same direction. So instead of being right, okay, you would have another set of these, another set of these, where all the ours would turn to Els and the Els would turn into ours. And then you have another set of those, right? Where the, um, with ours turning to s is, well, straight and the streets turn into rights. So basically, what you have is three times this set right here for a total of 18 outcomes. And again, if you want to actually make the list, that's how you do it. You just copy this list down, but interchanged R l's and the ours with B s is so that we have to right turn a set where there's two right turns and set where there's two left turns and said there's two straight. Okay, um, last list the outcomes of for these different things from All right. So we've got, um, de prime, which is the compliment of D, which is like, the anti d. All right, so yeah, D is the anti this direction. Okay, so, um, that would be the anti d with me. Either they're gonna all three go in the same direction, or all three will be different. Okay, All three go in different directions. Okay, so this would be the union of problem A and problem be so ace problem. Okay, so here's the idea. Right. So if they're not going to of them, are not going in the same direction. Which is that Todd D your tour? Not what that means, either. Either they're all going in different directions or they're all going the same direction. And we did those two problems already. So this in this so b list, eh? Along with a list B. Okay, Now, the next one is what's see union D. That means put see together with D. C's out together with these outcomes. Okay, So see would be too out of the three turn. Right. And we're putting that together where, um, two vehicles go in the same direction. So, actually, um, two vehicles going in the same direction two vehicles on the same direction is a bigger set than two vehicles turning right. So this is actually a subset of the other. So the answer to that is just gonna be the same list as D. Okay, It's gonna be the same list of D and then finally see intersect. E means that things that C and D have in common. So what? These have in common with the ones who were interchange them. And so the sea partisan both lists. Okay, so this part is again The sea is a subset of D. So it's going to be the overlap years of this one's gonna be C.