If two dice are rolled, what is the probability that the sum is four, or doubles are rolled?
Well first, let’s just remember that a dice is a numbered cube, and on each of the six faces is one of the numbers one to six. Now when you roll a dice, it lands so that one of the faces is facing upwards. And what we’re talking about is the number that faces upwards is the number that we are using.
Now because a fair dice is a perfect cube, all of the six numbers — one, two, three, four, five, and six — are equally likely to land face side up when you roll the dice. So that’s their background assumptions that we’re making.
Now the question says that we’re gonna be rolling two of these dice and we’re going to be adding up their scores, so the numbers facing upwards, and we wanna know that the probability that their sum is four or that doubles, or in other words the- the same number lands face up on both of the two dice. Well first of all, let’s just draw out a quick possibility space diagram.
Now let’s imagine we’ve got a pink dice and a blue dice, and there are 36 possible things that can happen. And the thing is because they’re fair dice here, all the numbers are equally likely to come up on each dice. All of these 36 outcomes are equally likely to occur.
Now that’s quite important to this problem, so we can get a one on the pink dice and a one on the blue dice. And if that happened, then the sum of those two would be two — one plus one is two. We could get a two on the pink dice and a one on the blue dice, so adding those two numbers together will give us three, and so on.
So this possibility space diagram maps all 36 different possibilities for things that could happen when you roll two dice and add together the scores. And we’re interested in, what is the probability that the sum is four?
Well there are three ways of that happening. So we could get a three on the pink dice and a one on the blue dice, or we could get a two on each dice, or we could get a one on the pink dice and a three on the blue dice.
So if the question had simply been what’s the probability that we get the sum of four when we roll these two dice, then there are 36 equally likely outcomes, three of which involve a four being the sum of the two numbers, so the probability would be three out of 36, but that’s not the question.
So let’s look at the other bit: What’s the probability that we get doubles being rolled? So the same number on each dice, what if I got a one on the pink and a one on the blue? That’d be this- this case here. If I had a two on the pink and a two on the blue, it would be that case there, and a three on each, four on each, five on each, and a six on each.
So the probability if this had been the question that we just get a double, in other words the same number on each dice, there are six ways of that occurring out of the 36 equally likely outcomes. So that answer would be six out of 36.
But the question asks, what’s the probability that we get a sum of four or the doubles are rolled? So it’s any of those situations. So what’s the probability that w- any one of those things that we’ve circled? Well how many different items have we circled? We’ve circled one, two, three, four, five, six, seven, eight different numbers.
So the probability that we get a sum of four or a double is eight out of 36. And if we wanted to, we could simplify that fraction. Eight out of 36 is the same as four out of 18, which is also the same as two out of nine.
Now all three of those are correct answers. You don’t have to simplify your answers in probability, but some people like to. Now it’s just worth mentioning before we go that we did circle this four here, getting two on both dice twice, and we didn’t double-count that particular outcome just because it fits the bill, it’s- the sum is four, and it’s a double. It’s still only one of the possible outcomes, so we only counted it once.