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Problem 77 Hard Difficulty

A friend who lives in Los Angeles makes frequent consulting trips to Washington, D.C.; 50$\%$ of the time she travels on airline $\# 1,30 \%$ of the time on airline $\# 2,$ and the remaining 20$\%$ of the
time on airline $\# 3 .$ For airline $\# 1,$ flights are late into D.C. 30$\%$ of the time and late into L. 10$\%$ of the time. For airline $\# 2,$ these percentages are 25$\%$ and $20 \%,$ whereas for airline $\# 3$ the percentages are 40$\%$ and 25$\%$ . If we learn that on a particular trip she arrived late at exactly one of the two destinations, what are the posterior probabilities of having flown on airlines $\# 1, \# 2,$ and $\# 3 ?$ Assume that the chance of a late arrival in L. A. is unaffected by what happens on the flight to D.C. [Hint: From the tip of each first-generation branch on a tree diagram, draw three second-
generation branches labeled, respectively, 0 late, 1 late, and 2 late. $]$

Answer

.4657 for airline #1, .2877 for airline #2, .2466 for airline #3

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Video Transcript

All right. So we're given some information about a friend of ours who likes going between Los Angeles and Washington, D. C. We're given three airlines. The probability that that friend uses each airline and the probability that they're late pulling either out of Washington see or into Los Angeles. So we have a hint that recommends that we set up a probability tree in a certain way. So we're gonna set up that probability tree right now. All right. So let a one b airline one a to B. Airline to et cetera. All right, so the probability of a wanted 0.50 probability of a chooser 0.30 the probability of a 30.20 that's what we're given. All right. Now I'm gonna define a new variable l zero ah, one and all to And what these mean is as follows l zero means that there are no late flights on either end. Ah, one means there's exactly one late flight Whether that be in l. A or Washington D. C. One late flight and l two means both flights are late. Now we're given the statistics that ah, certain either landing or taking office late in each city, and we're gonna use that to our advantage. So we're gonna make this lake three pronged thing off the site of each of our branches. Right? First off, we're gonna find L, too. So we're going to say probability of L, too. Devon, a one were given that they're late into dese on airline one, flights are laid into D. C. At 30% of the times and laid into L. A. A 10% of the time. That means the probability of both occurring 0.3 times 0.1 0.3 Conversely, probability of no late flights on a one. It's gonna be one minus the probability the flights are late in D. C. So is there a 0.7 times one minus probability? They're late in Los Angeles. Is there a 0.9? This is 0.63 now for l one. I'm gonna shoot the multiplication for this because these are only three possibilities, which means the last possibility is just gonna be one minus that all of the probabilities. So the other probabilities out of 2.66 So we're gonna do one minus 0.66 which is 0.54 No, that's wrong. Is there a 0.34? So now that we've done that for a one, let's do it for a two. So probability L too given a Jew is gonna be were given that they relate 25% and 20% of the time. So they're playing to five. I'm 0.2, and that's 0.5 Conversely, we have the probability of no late flights into a two, which is 0.75 times 0.8 0.6. Finally, we have l one given a two, there's gonna be one minus 0.65 Just add the other two together and this is 0.35 Finally, we're going to the same for airline three. So l want l, too. Given a three, it's gonna be 40 and 25%. So 0.4 times 0.25 that's 0.1. We have a probability of no late flights on a three gonna be 0.6 times 0.75 which is 0.45 And finally we're gonna figure out what l ones probability is, So am I. One minus 0.55 0.45 All right, so from this we know that the probability of l one a one a given a 10.34 probability of l one given a two equals probability of 35 Sorry. 0.35 I don't know what I just said. The probability of l one given a three equals 0.45 Now that we know that what we need to find our probabilities of the reverse of this so a one given l one probability of a two given l one probability a three given L one. Before that, we need to find probably ability of al one in general. And we can use a lot of total probabilities for that. So where it's gonna be 0.5 times 0.34 plus 0.3 times 0.35 plus point to 10.45 So it's right that over here it's gonna be 0.5 times 0.45 waas 0.3 times zero point for making sure this is right. Okay, This is why we check these things Because that isn't right. It's supposed to be 35 and 34 huh? Yeah. 34 and 350 wow. That wasn't right. It all good thing I checked. Otherwise, that would been a little embarrassing. 34 35 plus 0.2 times 0.45 There we go. And if you compute this, this comes out to be 0.365 All right, let's use some base there. Um, so we This is the probability of a one times the probability of l one given a one all divided by the probability of l one, it's gonna be 0.50 point 34 Divide by 0.365 This ends up being 0.4657 around four decimal places. Same thing here. Probability of a two times the probability of l one. Given a two all over the probability of that one. So this is gonna be 0.3 times 0.35 all over 0.365 And this ends up being 0.2877 Finally, we have a threesome. Probability of a three. Have the probability of l one, given a three over the probability of al one. So this is gonna be, uh, 0.2 times 0.45 all over 0.365 and this becomes zero point 2466 and there you have it.