A large operator of timeshare complexes requires anyone interested in making a purchase to first visit the site of interest. Historical data indicates that 20$\%$ of all potential purchasers select a day visit, 50$\%$ choose a one-night visit, and 30$\%$ opt for a two-night visit. In addition, 10$\%$ of day visitors ultimately make a purchase, 30$\%$ of night visitors buy a unit, and 20$\%$ of those visiting for two nights decide to buy. Suppose a visitor is randomly selected and found to have bought a timeshare. How likely is it that this person made a day visit? A one-night visit? A two-night visit?
we're giving this timeshare complex and just the proportion of those who visit and the proportion of those who ends up making a purchase. 20% of the people purchasing Earth Sorry, 20% of the people who visit she was a day visit 50%. She's a one night and the last 30% are two night visits were also given that the probability that somebody purchases given that they did a day visit, it's 10% probably that they purchased, given that it was a one night visit is 30% and the probability that they purchased given there were two night visit is 20%. So we're supposed to find the probability at their day they visited during the day, given that they purchased probability that they stayed for one night, given that they purchased the probability that they were hello to night visitor, given that they purchased. Now, as you can probably guess, all three of these require based them to some degree. But in order to find out, we need our probability of purchase week, which we can compute using our log total probability. It's going to be probability day times, the probability of purchase during the day, plus the probability of one night. Tens of probability they purchased, given it was a one night visit, plus the probability of a two night visit times the probability they purchased, given it was a two night visit, we plug all this into a calculator. You'll get our total probability that the purchase, which ends up being 0.23 All right, now, let's start using some base down, so this will be the probability. But the day purchased are sorry probability that they visited during the day, times the probability they purchased skin, that they were a day visitor, all divided by the probability of purchase, which is 0.20 time, 0.10 all over 0.23 And this comes out too. 0.0 87 All right, we're gonna do the same thing here. Next up seems last. All right. Uh, I'm starting to think I did not plan spacing out all these fractions well enough. But whatever will move on the probability of a one night purchases 0.50 Multiply that by the purchase. Given that there were one night visitor and then we divide that by the probability of purchase. And this comes out to 0.652 All right, I'm gonna preemptively move everything up because I feel like Oops. Ah, yeah, Because I feel like I did not plan this through. Right. There we go. Same old, same old. Same setup. It's can be 0.30 temp, 0.20 all divided by 0.23 This ends up being 0.261 and there you have it.