 Hi, this is Dr. Don. I have a problem out of Chapter 4 on discrete distributions. This particular problem requires the use of the geometric distribution. I've got another video in which I talk about how to decide which distribution to use, but just quickly, when you have a problem that talks about the first event happening, the first event happening in X number of trials, that is a geometric distribution problem. Here we're told that the manufacturer, the glass manufacturer says that one in every 200 items produced is warped, one out of 200 that gives you the probability of an item being warped. And we're asked to find, first of all, the first warp glass is the 12th item, the probability of that, the probability of the first warped item is the first, second, or third, and the probability that none of the first 10 items are defective. So we're going to do this using stat crunch. Okay, I have stat crunch open, and as we usually do, we go to stat, this time to calculators, and we look down, we find geometric, and we see we have two options, the number of failures or the number of trials. Well, in our problem, we're talking about the first event happening in the 12th trial, the 12th item. So we want to use the trials version of the calculator, and we bring it up, we need to enter the probability, and here we've just got one out of 200, and we don't have to calculate that, we can just put in one divided by 200, and that gives us our probability. The first question is the probability, the first warp glass item is the 12th item produced, so we click on our drop down and select equal, and we put in 12, because we want the first event that is successful to be the 12th event, and that would be .0047, which rounds to .005, which of course is the answer they have up here. Part B is the probability that the first success, the first event, the first warped item, is either the first, second, or third. So we go back to our calculator, and here we're saying that the event is either the first, second, or the third, so we put less than or equal three, because that would give us one, two, or three, and click compute, and that gives us .0149, which rounds to .015, which is the probability they want. Part C is none of the first 10 items are defective, well if none of the first 10 items are defective, that means our event could happen after the 11th item, the 11th trial. So we want to put greater than or equal 11, and that gives us a probability that the first event, the first success, which in our case is a warped glass item, occurs on trial 11 or greater, and that's .951, which is the answer they want here. Just for fun, let's look at the failure side of the calculator. I'm going back to stat, calculators, geometric failures, and bring up that calculator, and we're going to do our part C again. If none of the first 10 items are defective, that means that we have at least 10 failures, correct? If none of the 10 are defective, that means we don't have any successes for at least the first 10, so we need to go into our calculator and use the greater than or equal 10, so we want 10 or more failures, and again we need one divided by 200 for our probability, click compute, and that gives us the same probability .951, so we can either do it as 10 or more failures or 11 or more successes, so I hope this helps.