Below is a list of all the possible outcomes
in the experiment of rolling two dice.
Notice how we have a total of
six times six, or 36 possible outcomes.
We're asked to use the table to determine
the following probabilities.
We're first asked to determine the probability
the sum is odd.
Remember, odd numbers are numbers like one, three,
five, seven, nine and so on.
In order for a sum to be odd, one number must be even
and one number must be odd.
So you might recognize half of the outcomes
will have an odd sum.
Let's go ahead and circle them.
In the first row, one and two,
one and four, and one and six have an odd sum.
In the second row, two and one, two and three
and two and five have an odd sum.
In the third row, three and two, three and four,
and three and six have an odd sum.
In the fourth row, four and one, four and three,
and four and five have an odd sum.
In the fifth row, five and two, five and four,
and five and six have an odd sum.
In the last row, six and one, six and three,
and six and five have an odd sum.
There are 18 favorable outcomes and have a total of
36 possible outcomes.
The probability the sum is odd
is 18-36, which simplifies to one half,
which is a decimal is zero point five,
which is equal to 50%.
Next we're asked to find the probability the sum is five.
So going back to the table,
notice how we have a sum of five along this diagonal here.
Four and one have a sum of five,
three and two have a sum of five,
two and three have a sum of five,
and so do one and four.
So there are four favorable outcomes
out of a total of 36 possible outcomes.
The probability the sum is five is 436,
which does simplify.
Greatest common factor of 436 is four.
Simplify we divide both by four.
Which is equal to one ninth.
And let's find the decimal and percentage as well.
One divided by nine is point one repeating.
To four decimal places
this will round to zero point one one one one,
which is equal to 11.11 percent.
Next we're asked to find the probability the sum is seven
and at least one of the die is a one.
So both of these must be true
in order to be a favorable outcome.
Notice along this diagonal here the sum is seven.
But only the six and one and one and six have a sum
of seven and where the die shows a one.
And therefore there are only two favorable outcomes
out of a total of 36 possible outcomes.
The probability the sum is seven and at least one of the
die is a one, is 236.
Again, we don't count five and two, four and three,
three and four, and two and five as favorable outcomes
because while the sum is seven
at least one of the die is not a one.
So the probability is 236, which does simplify
dividing both by two
the simplified probability is one eighteenth.
Which is the decimal to four decimal places is approximately
zero point zero five, five six.
Which is equal to five point five six percent.
And for the last example we're asked to find the probability
the sum is seven or least one of the die is one.
So because of the or,
if either is true we have a favorable outcome.
Let's first circle the outcomes where the sum is seven,
which would be this diagonal here.
Six and one, five and two, four and three,
three and four, two and five and one and six,
all have a sum of seven.
And now let's circle the remaining favorable outcomes
where at least one of the die is one.
And one and one does have at least one die a one,
so one and one is favorable, so is one and two,
one and three, one and four, and one and five.
And then going down this column, two and one,
three and one, four and one, five and one
are also favorable outcomes.
So these are all the favorable outcomes where
the sum is seven or at least one of the die is one.
Let's count these favorable outcomes.
We have six in the first row,
and then going down the first column we have seven,
eight, nine, 10, 11.
And then 12, 13, 14, 15.
There are 15 favorable outcomes out of a total of 36
possible outcomes, the probability the sum is seven
or at least one of the die is one, is 15-36.
This does simplify.
15 and 36 share a common factor of three.
Simplifying we have five twelfths.
Which to four decimal places is approximately
zero point four one, six seven.
Which is 41.67 percent.
I hope you found this helpful.