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## The geometric distribution

# Cumulative geometric probability (less than a value)

AP.STATS:

UNC‑3 (EU)

, UNC‑3.E (LO)

, UNC‑3.E.2 (EK)

## Video transcript

- [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.

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