Layla and Michael each roll a six-sided dice numbered from one to six. They then find the product of the two numbers that they roll. (a) Work out the probability that the product is seven. (b) Work out the probability that the product is a number less than seven.
So we’re going to be working with probabilities. And for part a, we’re asked to work out the probability that the product is
seven. The six-sided dice, the faces or the sides are one through six. Layla rolls and then Michael rolls and then they find the product, meaning they
multiply the numbers together. And we want to know the probability that the product is seven.
Well, the only way to get a product of seven would be one times seven or seven times
one because seven is prime; it only has two factors: itself and one. So since we have to multiply by a seven in order to get seven, seven would have to be
one of the sides of the dice.
And there is no number seven. So the probability that this would happen will be zero because it can’t happen. It’s simply not possible.
Now, for part b, it says, “Work out the probability that the product is a number less
So, this part is very similar. But this time, the product isn’t seven. It’s a number that’s less than seven. So those numbers would be, so numbers that are less than seven are one, two, three,
four, five, and six.
So let’s list the ways to roll each of these numbers: one, two, three, four, five,
and six because these are the numbers that are less than seven. We can get one from one times one and that’s it.
Now, for two, we can take one times two or we can take two times one. It will make us think, “Well, this is the exact same thing.” Well, imagine, remember these are the rolls. So let’s say Layla goes first and she rolls a one and then Michael rolls and he gets
a two. That’s different than Layla first rolling the two and then Michael rolling the
For three, we can have one times three or three times one.
There’s a few more options for four. We can have one times four or four times one. And we could also have two times two.
For five, we can have one times five or five times one.
For six, we can have one times six or six times one. And then, there’s a few more. We can have two times three or three times two.
So how many ways are there to roll a product less than seven? Let’s count: one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13,
14. So the number of ways to roll a product less than seven will be 14.
Now to find a probability, we would need to take this number and put it out of, so
over, all the possible rolls. So again, we take the ways to roll a product less than seven all of those ways and
then put it out of all of the possible rolls.
So Layla could roll numbers one through six and Michael could roll numbers one
through six. So one possibility would be Layla rolling a one and Michael rolling a one, Layla
rolling a one and then Michael rolling a two, Layla rolling a one and then Michael
rolling a three, and so on. So there are six possibilities with Layla rolling the one.
Now, let’s say Layla rolled a two. After that, Michael could roll a one, a two, three, four, five, or six. So there are six possibilities. We could repeat this process with three. There will be six possibilities for three.
And if we would repeat this process for four, five, and six, we would get six, six,
and six. So altogether, adding those up, there will be 36 six total possibilities.
We also could have found this by multiplying, so the six numbers that Layla could
have gotten times the six numbers that Michael could have gotten. Six times six would give us 36.
So this probability would be 14 out of 36. However, we can reduce this. Both numbers are divisible by two. So we will get seven eighteenths.
So once again, the probability that the product is seven was zero and the probability
that the product is a number less than seven will be seven eighteenths.