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## Compound events and sample spaces

Current time:0:00Total duration:5:15

## Video transcript

Find the probability
of rolling doubles on two six-sided dice
numbered from 1 to 6. So when they're talking
about rolling doubles, they're just saying,
if I roll the two dice, I get the same number
on the top of both. So, for example, a 1
and a 1, that's doubles. A 2 and a 2, that is doubles. A 3 and a 3, a 4 and a 4,
a 5 and a 5, a 6 and a 6, all of those are
instances of doubles. So the event in question
is rolling doubles on two six-sided dice
numbered from 1 to 6. So let's think about all
of the possible outcomes. Or another way to
think about it, let's think about the
sample space here. So what can we roll
on the first die. So let me write this
as die number 1. What are the possible rolls? Well, they're
numbered from 1 to 6. It's a six-sided die, so I can
get a 1, a 2, a 3, a 4, a 5, or a 6. Now let's think about the
second die, so die number 2. Well, exact same thing. I could get a 1, a 2,
a 3, a 4, a 5, or a 6. Now, given these possible
outcomes for each of the die, we can now think of the
outcomes for both die. So, for example, in this--
let me draw a grid here just to make it a little bit neater. So let me draw a line there and
then a line right over there. Let me draw actually
several of these, just so that we could really
do this a little bit clearer. So let me draw a full grid. All right. And then let me draw the
vertical lines, only a few more left. There we go. Now, all of this top row,
these are the outcomes where I roll a 1
on the first die. So I roll a 1 on the first die. These are all of those outcomes. And this would be I run
a 1 on the second die, but I'll fill that in later. These are all of the
outcomes where I roll a 2 on the first die. This is where I roll
a 3 on the first die. 4-- I think you get the
idea-- on the first die. And then a 5 on
the first to die. And then finally, this last
row is all the outcomes where I roll a 6
on the first die. Now, we can go
through the columns, and this first column is where
we roll a 1 on the second die. This is where we roll
a 2 on the second die. So let's draw that out, write
it out, and fill in the chart. Here's where we roll
a 3 on the second die. This is a comma that I'm
doing between the two numbers. Here is where we have a 4. And then here is where
we roll a 5 on the second die, just filling this in. This last column is where we
roll a 6 on the second die. Now, every one of these
represents a possible outcome. This outcome is where we roll
a 1 on the first die and a 1 on the second die. This outcome is where we
roll a 3 on the first die, a 2 on the second die. This outcome is where we
roll a 4 on the first die and a 5 on the second die. And you can see here, there are
36 possible outcomes, 6 times 6 possible outcomes. Now, with this out of the way,
how many of these outcomes satisfy our criteria of rolling
doubles on two six-sided dice? How many of these outcomes
are essentially described by our event? Well, we see them right here. Doubles, well, that's rolling
a 1 and 1, that's a 2 and a 2, a 3 and a 3, a 4 and a 4, a
5 and a 5, and a 6 and a 6. So we have 1, 2, 3, 4, 5, 6
events satisfy this event, or are the outcomes that are
consistent with this event. Now given that, let's
answer our question. What is the probability
of rolling doubles on two six-sided die
numbered from 1 to 6? Well, the probability
is going to be equal to the number of outcomes
that satisfy our criteria, or the number of outcomes
for this event, which are 6-- we just figured
that out-- over the total-- I want to do that pink
color-- number of outcomes, over the size of
our sample space. So this right over here,
we have 36 total outcomes. So we have 36 outcomes,
and if you simplify this, 6/36 is the same thing as 1/6. So the probability
of rolling doubles on two six-sided dice
numbered from 1 to 6 is 1/6.