 Hi, folks. This is Dr. Don. I have a problem out of Chapter 3 on probabilities. And this particular problem is about finding conditional probabilities. We're given a table of information up here about men and women and whether or not they have savings put aside. And it is a 2x2 matrix with men and women on top and less than one month income, more than one month income on the vertical axis. We're going to do this using an Excel calculator or an Excel cheat sheet that I put together. I find more students missed this problem on quizzes and exams than any other by not really doing the math properly or setting up the calculations properly. So I built this calculator that will help both with the Continuity table we use to solve this, finding the conditional probabilities as well as simple probabilities, joint probabilities, ANB, the addition rule, ARB, and then of course conditional probabilities. Let's take a look at it. We start by clicking on the little blue rectangle there. And instead of opening to a clipboard, I mean, Excel, I'm going to copy to the clipboard again. Remember, as long as it's blue, we can right click and copy. And then I'm going to go over here to the worksheet that you can download. I'll give you the link. And I'm going to scroll over here to the right a bit and use this blue area that I call the scratch pad. And I'm going to paste my information in there. Now I do that because I mentioned the good folks in my stat lab give us this row total and column totals, but we don't need that and it won't fit in my cheat sheet. So I'm going to just copy the information we need, the labels for the rows and the columns and then the data Excel. And remember, we got the little cell there in the corner. I'm going to right click and copy that that I'm going to go over here to the main part of the calculator. I'm going to pick that little cell right there again and paste in the information. And as simple as that, we have all the answers. I've set this up so it will find the simple probabilities, the probability that that is a man is 0.498, probability woman 0.501. But what we want for this problem is the conditional probabilities. Let's go back over here and look at the problem they want. Find probability that a randomly selected worker has one month income or more set aside. Now that is just a simple probability. So we look in the simple probability section, the probability of one month income or more is 0.408. They wanted three decimal places, which gives you 0.481. Second one, given that a randomly selected worker is a male, find the probability worker has less than one month income. So if we state that using math format, you know, what is the probability of one month income given male? So we look over here in the table, look down in conditional probabilities, the probability one month income given men or male is 0.4615 or that rounds to 0.462, which is the answer they want over here. The third question, given that a randomly selected worker has one month or income or more, find that probably the work is female. So that's probably a female given one month's income or more. So we look over here, probability of women given one month's income or more is 0.442, which is the answer they want there. Part D asks, are the events having less than one month's savings and being male independent or dependent? Events are independent if the probability of one event occurring does not affect the probability of the other event occurring. So we can check to see if the probability of being male is different than the probability of being male given one month's savings. Using our math format, we would say probability of male equal probability of male given less than one month's savings. Looking in the calculator, we see that the probability of being a man being male is 0.4983. Looking down lower, we see that our conditional probability section, the probability of being male and having less than one month's income is 0.4430. 0.4430 and 0.4983 are not equal. Therefore, we have to conclude the events are dependent. Hope this helps. And if it does help, please consider subscribing to my YouTube channel, the stats files. Just click on the big red button.