Who did you know? The general wisdom is that you're supposed to go to the dentist twice a year. So that's interesting. Well as you study this table. So this table this this contingency table. This problem of a probability distribution, probability distribution. Yes. Um ranks respondents to the survey by age group and by how often or when was the last time that they had been to the dentist? So um that here we go. Let's go and jump into our problem. So it's supposed to represent the whole U. S. Population. So now we're going to ask we can take from the survey and generalize it to the U. S. Population. Say if a american adult is selected at random, what is the probability That they fall into one of these categories? We're gonna go ahead and convert them two as we go along. So the first one asks about our first row, what's the probability that randomly selected american adult has attended? The debt has been to the dentist in within the last six months, basically is how you would translate that. Right? So that's asking for that single variable, not a joint event. So, headway on out to the margin there and get that marginal probability associated with less than six months. So that's a 0.441 out there on the table. Alright. The next one is the probability that the person, the randomly selected american adult is not in the second age category. Okay, so the easiest way to do this is to take the totality of all the people and subtract off the marginal probability for the second age group. Second column of age groups. So that's a fair point 314. So if we do that calculation uh handy Dandy calculator real quick. We end up with 0.686. So you should be interpreting these as we go along. So the first one, you know, you make a sentence of that. You're going to say like 44.1%. There is there is a 44.1 chance that a randomly selected American citizen adult has been to the dentist in within the last six months. It's not bad. That's not bad. 44 chance. So yeah, if an American adult is selected at random that there is a 44 probability that they have been to the dentist within the last six months. So this one Would go, there's a 68.6 chance that a randomly selected american adult is within the age group of 45- 64. All right. That's just saying how old they are. Really has nothing to do with the dentist. Right. All right. So let's come to the next one. So, the next one is a joint event alright. The joint event that they are over 75 and that it's been five or more years since they've been to the dentist. Okay. So people who just basically don't go to the dentist anymore and are very old. What's the probability that a random select american citizen falls in that joint event category? So you're pulling that off, you're looking for the intersection of row five there and call them for Within that black box. And that would translate as 2.2 chance. So let's say if an American adult was selected at random, there's a 2.2 chance that they are over 75 and have not been to the dentist in more than five years, five or more years, say it correctly. Alright, onward. Now we get to our nice conditional probability is really important part of the question. Right. What's the probability that they've been to the dentist sometime between two and five years. And I'm switching up my notation here given That they are an adult under 44 years old. All right. So that you've been to the dentist into in the last 2-5 years, But you're an adult under 44 given given. Right. So, we got to look only at so in these when we have given a probability distribution table and ask are conditional. Were forced to do this formula where we have to do um Joint variation divided by the given. The probability is given. So we want the probability of T4 and a one divided by The probability of the given, which was just a one. All right. So look for that joint one at row four and column one. That's 10.7 Provided by that marginal probability at the bottom of row of column one, which is .5-6. All right, pull out the handy dandy calculator for these 24 755.5 to 6. It's And we get .133 or there's a 13.3 chance who 13.3 chance that are randomly selected American adults is Under 44 and has been to the dentist in the last 2, 5 years. All right. Last but not least. They take that same conditional, but swap it around. So now it's the condition conditional probability. What's probability that they are 18 to 44 years old given That they've been to the dentist in the last 2-4, 2, 5 years. Right. So this numerator is exactly the same for this calculator calculation, but the denominator now switches to that T. four row, so My numerator still .7, but my denominator becomes the marginal probability for roti four, which is 40.1 to 2. All right, So he is. The calculator gives me 25 7 Or 57.38%. So we can say, given that a randomly selected adult Has been to the dentist in the last 2 to 5 years. The probability that they are 18-44 is 57.38%. All right, enjoy.