Alright in this question were given that data about NFL players their years of experience in the columns, their weight categories in the rose and asked to calculate a bunch of probabilities Using the frequencies that are given. Remember the table You're given a joint frequencies right, represents a tally of players that meet those categories. We're going to take those numbers, divide them by our total and then get probabilities which will be numbers less than one. Okay, so always be paying attention to what kind of numbers you're using and what you're doing with them right now. Not all of the questions. All of the sub questions here are conditional. The first one is not it's a straightforward requests for probability. The questions if you choose randomly one of these 65 players, what is the probability that you will selected a rookie? Okay. So you want to know how many refuse total. So we're headed out here outside the box to the margin. Right? So in order to calculate the rookie mm probability the probability is that we have a rookie and we're gonna want to take all of the rookies. They're divided by the total. We'll have our handy dandy spreadsheet. Do that. Click on the 11 Divided by Our Killer 65. All right. And let's set this to have not too many decimal places. Right. There we go. That's better. four Decimal Places. There's plenty. Alright, the next question is About the weight. We want to know the probability that the weight is under 200. Right? Good. Less than 200. So that's again one simple category. Not a joint event. And so we want to go out to the margins right? To all of the players that way under 200 and take that number That eight divided by the total population of 65. Alright, so that's just straightforward kind of, you know, very it probability questions. Now, we start getting to the conditional probability questions. Okay, So now we're being asked what's the probability? Right? But you have a rookie but not out of all 65 players. We wanted to know what they're probably having a rookie Assuming or given that we're only considering players the way under 200. Right. Just looking at the category of those who weigh under 200. Right, so just looking at this category, what's the probability of getting a rookie? That's your column once earlier. That intersection of column 1? En route one. Okay, so um the nominator For this fact for this probability will no longer be 65. Because I'm not considering that I could draw from all the players, I'm only considering that I could draw from those in the lowest weight class, Which is the eight. And the number of rookies in that lowest weight class is just three. So this calculation is the three over the eight. Mhm. So in the next one, We've got ways under £2 or under £200.. Given I use a race, there's the same relationships. The numerator is going to be the same because I'm still looking at a player That's both under 200 and a rookie, so it's still going to use that three. But the denominator is different because now we are going to assume that we're looking at all the rookies, And I just want to know which one is under 200. Right? So that probability expression looks backward. I want to know if he's a lightweight, given that he's a first year player. Let's see how that a little backwards. We're writing up to symbols. Alright, so, again, someone looking column C. Because given that he's a rookie, so we're going to choose that joint event divided by the total of the rookies. Okay, so there's your probability for so into question E. It's just a matter of moving all those decimal places to to the right. All right. Watch what happens when you just click the button there, Right. And they just move that decimal places two places to the right and put the percent sign on. All right. So you can if they interpret expects sentences, you can say things like a third of all of the lowest weight class. Um, players, our first year players, that kind of comment that you can inmate or approximately Or be precisely say 37.5%. All right, that's all for this question. Hope you enjoyed it.