Geometric random variables introduction

- [Narrator] So I have two,
different random variables here.

And what I wanna do is think about

what type of random variables they are.

So this first random variable, x,

is equal to the number of sixes

after 12 rolls of a fair die.

Well this looks pretty much like

a binomial random variable.

In fact, I'm pretty confident it is

a binomial random variable and we

can just go down the checklist.

The outcome of each trial can be

a success or failure.

So, trial outcome success or failure.

It's either gonna go either way.

The result of each trial is independent

from the other one.

Whether I get a six on the third trial

is independent of whether I got

a six on the first or the second trial.

So result, let me write this,

trial, I'll just do a shorthand trial,

results independent, independent,

that's an important condition.

Let's see, there are a
fixed number of trials.

Fixed number of trials.

In this case we're gonna have 12 trials.

And then the last one is, we have

the same probability on each trial.

Same probability of success

probability on each trial.

So yes indeed, this met
all of the conditions

for being a binomial,
binomial random variable.

And this was all just a little

bit of review about things that

we have talked about in other videos.

But what about this thing
in the salmon color?

The random variable y.

So this says the number of rolls

until we get a six on a fair die.

So this one strikes us as
a little bit different.

But let's see where it
is actually different.

So, does it meet that the trial outcomes

that there's a clear success
or failure for each trial?

Well yeah, we're just gonna keep rolling.

So each time we roll, it's a trial.

And success is when we get a six.

Failure is when we don't get a six.

So the outcome of each trial can

be classified as either
a success or a failure.

So it meets, maybe I'll put the checks

right over here, it meets
this first constraint.

Are the results of each trial independent?

Well whether I get a six on the first roll

or the second roll, or the third roll,

or the fourth roll, or the third roll,

the probabilities shouldn't be dependent

on whether I did or didn't
get a six on a previous roll.

So, we have the independents.

And we also have the same probability

of success on each trial.

In every case it's a 1/6 probability that

I get a six, so this stays constant.

And I skipped this third
condition for a reason.

Because we clearly don't have
a fixed number of trials.

Over here we could roll 50
times until we get a six.

The probability that we'd have

to roll 50 times is very low.

But we might have to roll 500 times

in order to get a six.

In fact, think about what
the minimum value of y is

and what the maximum value of y is.

So the minimum value that
this random variable can take,

I'll just call it min y, is equal to what?

Well, it's gonna take at least one roll.

So that's the minimum value.

But what is the maximum value for y?

And I'll let you think about that.

I'll assumed you thought about
it, if you paused the video.

Well, there is no max value.

You can't say, "Oh it's a billion."

Because there's some probability

that it might take a
billion and one rolls.

It is a very, very, very,
very, very, very small

probability, but there's some probability.

It could take a Google
rolls, a Google plex rolls.

So you can imagine where this is going.

So this type of random variable,

where it meets a lot of the constraints

of a binomial random variable.

Each trial has a clear
success or failure outcome.

The probability of success
on each trial is constant.

The trial results are
independent of each other.

But we don't have a
fixed number of trials.

In fact, it's a situation, we're saying,

"How many trials do we need to get,

"to we need to have until we get success?"

Maybe that's a general way of framing

this type of random variable.

How many trials until success?

While the binomial random variable was,

how many trials, or how many successes,

I should say, how many successes in

finite number of trials?

So if you see this general form

and it meets these conditions, you can

feel good it's a binomial random variable.

But if we're meeting this condition,

clear success or failure outcome,

independent trials, constant probability,

but we're not talking about the successes

in a finite number of trials.

We're talking about how
many trials until success?

Then this type of random variable

is called a geometric random variable.

And we will see why, in future videos

it is called geometric.

Because the math that involves

the probabilities of various outcomes

looks a lot like geometric growth,

or geometric sequences and series

that we look at in other
types of mathematics.

And in case I forgot to mention,

the reason why they're called binomial

random variables is because when you

think about the probabilities
of different outcomes,

you have these things called
binomial coefficients,

based on combinatorics.

And those come out of things like

Pascal's Triangle and when you take

a binomial to ever increasing powers.

So that's where those words come from.

But in the next few videos, the important

thing is to recognize the
difference between the two.

And then we're gonna start thinking

about how do we deal with
geometric random variables.