A fair die is rolled twice.
A fair die is a six sided die as shown here on the right.
The sample space for one roll of a fair die
is 1, 2, 3, 4, 5, 6,
which means these are the possible outcomes
when rolling a fair die once.
But again, we are told to roll a fair die twice
and the first question is
are the two rolls independent events?
Independent events are events where an outcome of one event
is not affected by the outcome of another.
And dependent events are events where an outcome
of one event is affected by the outcome of another.
So in our case, the outcome of the second roll
is not affected by the outcome of the first roll
and therefore the two rolls are independent events.
And therefore we answer yes for part A.
If A and B are independent events,
then the probability of both A and B occurring
is equal to the probability A times
the probability of B.
Next, we're asked to find the probability
that both rolls show a six.
We indicate this as the probability
of rolling a six on the first roll
and a six on the second roll.
And because these are independent events,
this is equal to the probability of rolling a six
times the probability of rolling a six.
So looking at the sample space,
we know we have six possible outcomes,
only one of which is favorable for rolling a six.
The probability of rolling a six is 1/6
and therefore the probability of rolling a six on both rolls
is equal to 1/6 times 1/6, which equals 1/36.
And we can verify this if we take a look
at the sample space below for rolling a fair die twice.
Notice how there are six times six or 36 possible outcomes,
only one of which is favorable, where both rolls are a six,
which does verify the probability
of rolling a six on the first roll
and a six on the second roll is 1/36.
But let's also show the probability
as a decimal and percentage.
To convert the fraction to a decimal, we divide one by 36,
which to four decimal places is 0.0278,
where as the percentage is 2.78%.
Just keep in mind the fraction is exact,
these two have been rounded.
Next, we're asked to find the probability
the first roll is an even number
and the second roll is a one.
So we want the probability of an even on the first roll
and a one on the second roll.
And because we have two independent events,
this is equal to the probability of rolling an even
times the probability of rolling a one.
Going back to the sample space of rolling one fair die,
there are three favorable outcomes rolling an even number,
which are two, four, and six.
The probability of rolling even is equal to 3/6 or 1/2.
Let's go ahead and write 3/6
and then times the probability of rolling a one
because there's one favorable outcome
out of a total of a total of six,
the probability of rolling one is 1/6.
Before determining this product,
notice how three and six share a common factor of three.
There is one three in three and two threes in six.
Now we have 1/6 times 1/2, which is equal to 1/12.
The probability of rolling an even number on the first roll
and the number one on the second roll is 1/12.
Again, if we go back and take a look at the sample space
for two rolls of a fair die,
let's verify this probability.
Notice the first column is where we find
the second roll is a one
and therefore the outcomes that show
an even number on the first roll
and a one on the second roll are
2 comma 1, 4 comma 1, and six comma 1.
There are three favorable outcomes
out of a total 36 outcomes,
giving us the probability of an even number
on the first roll and a one on the second roll
is equal to 3/36, which does simplify to 1/12.
Again, let us show the probability
as a decimal and percentage.
One divided by 12 to four decimal places is 0.0833,
which is equal to 8.33%.
And for the last example, we're asked to find
the probability both rolls show an odd number,
which is the probability of an odd on the first roll
and an odd on the second roll.
Because we have independent events,
this is equal to the probability
of rolling an odd times the probability of rolling an odd.
And the odd numbers from one through six are
one, three, and five.
There are three favorable outcomes
out of a total of six outcomes.
The probability of rolling an odd is 3/6 or 1/2.
We'll start with 3/6 and then times
the probability of rolling an odd
which we already know is 3/6.
3/6 simplifies to 1/2, giving us 1/2 times 1/2,
which is equal to 1/4 as a fraction,
0.25 as a decimal, and 25% as a percent.
And again, let's go ahead verify this
by looking at the sample space for two rolls of a fair die.
Notice only rows one, three, and five
show an odd number on the first roll
and therefore we can ignore rows two, four, and six
because they show an even number on the first roll.
So in the first row, both rolls are odd when we have
one comma one, one comma three, and one comma five.
Looking at row three, both rolls are odd when we have
three comma one, three comma three, and three comma five.
In the fifth row, both rolls are odd when we have
five comma one, five comma three, and five comma five.
We have nine favorable outcomes
out of a total of 36 outcomes,
which gives the probability of an odd
on both rolls as 9/36, which does simply to 1/4.
I hope you found this helpful.