Two dice are thrown together. Find the probability that the
numbers obtained have i) an even sum or ii) an even product.
We know that the probability of an
event equals the number of ways that the event can occur over all possible
outcomes. Our first option wants to know the
probability that we would have an even sum. The first die could turn up
anything from one to six, and so could the second one. And we need to consider the sum of
all of these cases.
One plus one is two. One plus two is three. One plus three is four. One plus four is five. One plus five is six and then
seven. We will continue this, finding the
sum of each two values, until the table is full. Remember that we’re considering the
places where we would have an even sum. Now, let’s circle all the places
where there was an even sum. In total, there are 18 ways an even
sum occurs. And there are 36 possible
outcomes. The numerator and the denominator
are both divisible by 18. 18 divided by 18 is one. 36 divided by 18 is two. The probability that we would get
an even sum is one-half. You can expect that if two dice are
thrown together, half of the time, they will have an even sum.
Moving on to part two, we need to
consider the case that we find an even product. We can use the same strategy. The first die has options one
through six, as does the second. And we need to consider the product
of each pair. One times one is one. One times two is two. One times three is three. One times four is four. And we’ll continue to multiply each
pair until the table is full. After we find all the products,
we’ll circle the ones that are even.
The number of ways that an even
product can occur is 27. All possible outcomes, there are 36
possible outcomes. Both the numerator and the
denominator of this probability are divisible by nine. 27 divided by nine is three. 36 divided by nine is four. The probability that you would roll
an even product if two dice are thrown is three-fourths.