Find the probability of rolling doubles on two
six-sided dice, numbered from 1 to 6.
So let's just think about it.
Let's think about all of the outcomes, all of the possible
outcomes, and then what fraction of those outcomes
represent us getting doubles.
We want to roll doubles.
So let's make a little table here that'll give us all of
our outcomes.
You have 6 outcomes on the first dice, or the first die I
should say.
So 1, 2, 3, 4, 5, 6.
So let's call this the first die.
And then the second die, there are 6 outcomes.
So 1, 2, 3, 4, 5, 6.
This is the second die.
So all of the different possible outcomes are going to
be some-- you're going to get an outcome-- so you could get
a 3 over here.
And then when you get a 3 on the second die, you might have
gotten a 1, 2, 3, 4, 5, or 6 on the first die.
Or you might have gotten a 5 on the first die and then you
could get a 1, 2 3, 4, 5, or 6 on the second die.
So if I draw a grid here.
If I were to draw a grid here, let me do my best to draw a
quick grid.
Just draw it really fast. So let me just draw this grid
right here.
Each cell of this grid, each box in this grid, would
represent an outcome.
All of the possible outcomes.
Let me finish drawing it.
So there you go.
And so just to get an understanding of what I mean
by it represents an outcome.
This cell right here would represent getting a 4 on your
first die and a 2 on your second die.
That's the outcome it represents.
It's one of the potential outcomes.
Now how many total outcomes are there?
Well you could look at the area of this right here.
The area of this is going to be 6 times 6.
You have 36 possible outcomes.
So 36 possible outcomes.
And it makes sense.
You can think about the first die.
There's 6 outcomes.
And then for each of those 6 outcomes, you have 6 possible
outcomes for the second die.
So 6 times 6.
Or you can think of in this grid, it has
exactly 36 cells in it.
Now what are the outcomes that satisfy our conditions?
That entail us getting doubles?
What are the outcomes there?
Well we could get a 1.
If we get a 1 on the first die, we have to get a 1 on the
second die.
If we get a 2 on the first die, we have to get a 2 on the
second die.
If we get a 3 on the first die, we have to get a 3 on the
second die.
I think you see where this is going.
4 and 4.
5 and 5.
And then 6 and 6.
So how many of these outcomes involve us getting double?
Well, it's 1, 2, 3, 4, 5, 6 outcomes.
So there's 6 outcomes with doubles.
So if someone asks you, what's the
probability of getting doubles?
The probability of getting doubles when you rolled two
six-sided dice, numbered from 1 to 6.
It's going to be equal to the number of outcomes that give
us doubles-- so it's going to be equal to 6-- over the
number of the total number of possible outcomes.
Over 36.
And 6 over 36 is 1/6.
Now I made sure that you kind of understood that look, these
are all the possible outcomes.
And 6 of them satisfy.
So 6 of the 36.
So there's a 1/6 probability of rolling doubles.
But another way you could have thought about it is, you know
what, let's just roll the first die.
It's going to have some number on it, right?
The first die is going to have some number on it.
Now in order for me to have doubles, the second die has to
have that same exact number on it, right?
So I don't care what the first die is.
It could be a 3, it could be a 4, it could be a 5.
I don't know what that is.
But after I've rolled that first die, in order to get
doubles, what the second die roll is-- they might be almost
simultaneous-- but what the second die rolls matters.
And so whatever number I have on the first die, there's a
1/6 chance that the second die is going to have that same
number on it.
So that's another way of reasoning that
you have a 1/6 chance.
The first die does anything it wants, but there's only a 1 in
6 chance that the second die is going to have the same
thing as the first die.
Either way, you get to the same 1/6 answer.