Video Transcript
Two dice are rolled to give a pair of numbers. Given that both numbers are greater than one, what is the probability that they are
both equal to two?
Now we could apply a little bit of logic to calculate the probability here. However, the phrase “given that” tells us that we’re actually working with a
conditional probability. And in this case, we actually have a formula that we can use. The formula tells us that the probability of an event 𝐴 occurring given that an
event 𝐵 has already occurred is equal to the probability of 𝐴 intersection 𝐵 over
the probability of 𝐵. Remember, this vertical lines simply means given that. And this symbol that looks a little bit like a lowercase n is the intersection. When we talk about the intersection, we’re talking about 𝐴 and 𝐵 occurring.
So we’re going to define our two events to start with. We’re going to say that 𝐴 is the event that both dice are equal to two. Then 𝐵 is the event that both numbers are greater than one. And that’s great because the probability of 𝐴 given 𝐵 is now the probability that
both dice are equal to two given that both numbers are greater than one. We now need to calculate the probability of 𝐴 and 𝐵 or 𝐴 intersection 𝐵 and the
probability of 𝐵.
And so we’re going to recall that for two independent events, let’s now all those 𝐶
and 𝐷, the probability of 𝐶 intersection 𝐷 occurring is equal to the probability
of 𝐶 times the probability of 𝐷. Now we chose 𝐶 and 𝐷 very specifically here. The outcomes we’re now interested in is the score we get on each individual dice. Note that the score on one dice doesn’t affect the score on the other. And so rolling each dice is an independent event.
Let’s begin with the probability of 𝐵. We know that this is both numbers being greater than one. Well, that means each dice could have the numbers two, three, four, five, or six on
them. The probability that the score on one dice is greater than one then is
five-sixths. So the the probability that the score is greater than one on both dice is five-sixths
times five-sixths. When we multiply fractions, we simply multiply their two denominators and their two
numerators. So we get 25 over 36.
But what about the probability of 𝐴 intersection 𝐵? This is the probability that both dice are equal to two and that both numbers are
greater than one. Well, two is always greater than one. And so actually the probability of 𝐴 intersection 𝐵 here is the same as the
probability of 𝐴. The probability that the number on one dice is equal to two is one-sixth. So the probability that both dice are equal to two is one-sixth times one-sixth. And that’s equal to one over 36. And we’re now ready to use the conditional probability formula.
The probability of 𝐴 given that 𝐵 has already occurred then is one over 36 all
divided 25 over 36. We know that to divide by a fraction, we simply multiply by the reciprocal of that
fraction. So we get one over 36 times 36 over 25. Then we spot that we can cross cancel by dividing through by a common factor of
36. We then have one over one times one over 25. And that’s equal to one over 25. Given that both numbers are greater than one then, the probability that they’re both
equal to two is one out of 25.
