Cumulative geometric probability (less than a value)

- [Instructor] Lilyana runs a cake decorating business, for which 10% of her orders come over the telephone. Let C be the number of cake orders Lilyana receives in a month until she first gets an order over the telephone. Assume the method of placing each cake order is independent. So C, if we assume a few things, is a classic geometric random variable. What tells us that? Well, a giveaway is that we're gonna keep doing these independent trials, where the probability of success is constant, and there's a clear success. A telephone order in this case is a success. The probability is 10% of it happening. And we're gonna keep doing it until we get a success. So classic geometric random variable. Now they ask us, find the probability, the probability, that it takes fewer than five orders for Lilyana to get her first telephone order of the month. So it's really the probability that C is less than five. So like always, pause this video and have a go at it. And even if you struggle with it, that's better. Your brain will be more primed for the actual solution that we can go through together. Alright. (chuckles) So I'm assuming you've had a go at it. So there's a couple of ways to approach it. You could say, well, look, this is just gonna be the probability that C is equal to one, plus the probability that C is equal to two, plus the probability that C is equal to three, plus the probability that C is equal to four. and we can calculate it this way. What is the probability that C equals one? Well, it's the probability that her very first order is a telephone order. And so we'll have 0.1. What's the probability that C equals two? Well, it's the probability that her first order is not a telephone order. So it's one minus 10%. There's a 90% chance it's not a telephone order, and that her second order is a telephone order. What about the probability C equals three? Well, her first two orders would not be telephone orders, and her third order would be one. And then C equals four? Well, her first three orders would not be telephone orders, and her fourth one would. And we could get a calculator maybe, and add all of these things up, and we would actually get the answer. But you're probably wondering, well, this is kind of hairy to type into a calculator. Maybe there is an easier way to tackle this. And indeed, there is. So think about it. The probability that C is less than five, that's the same thing as one minus the probability that we don't have a telephone order in the first four. One minus the probability that no telephone order in first four orders. So what's this? Well, 'cause this is just saying, what's the probability we do have an order in the first four? So it's the same thing as one minus the probability that we don't have an order in the first four. And this is pretty straightforward to calculate. So this is going to be equal to one minus, and let me do this in another color so we know what I'm referring to. So what's the probability that we have no telephone orders in the first four orders? Well, the probability on a given order that you don't have a telephone order is 0.9. And then if that has to be true for the first four, well, it's gonna be 0.9 times 0.9 times 09 times 0.9, or 0.9 to the fourth power. So this a lot easier to calculate, so let's do that. Let's get a calculator out. Alright, so let me just take .9 to the fourth power is equal to, and then let me subtract that from one. So let me make that a negative, and then let me add one to it. And we get, there you go, 0.3439. So this is equal to 0.3439. And we're done. That's the probability that it takes fewer than five orders for her to get her first telephone order of the month.