# Video: Pack 4 • Paper 1 • Question 3

Pack 4 • Paper 1 • Question 3

04:03

### Video Transcript

Vinnie has a spinner. There are four possible outcomes that the spinner could land on: orange, purple, brown, and blue. The probabilities of landing on orange and landing on blue are shown below. The probability of landing on purple is twice the probability of landing on brown. Complete the table.

So if we’re trying to solve this problem, what we need to do is find out the probability of purple and the probability of brown. So a bit of information that we have is that the property of landing on purple is twice the probability of landing on brown.

So what we can do to actually help us solve this problem is called the probability of brown 𝑥. So we’re saying the probability of brown is equal to 𝑥. And therefore, we can say that the probability of purple is gonna be equal to two 𝑥. And that’s because, as we said, the probability of landing on purple is twice the probability of landing on brown. So I’ve actually put those labels just below the actual table so we can see what we’re working with.

Okay, so now what do we do next? Well, what we know is that the probabilities sum to one. And this is because, actually, we have an exhausted set of outcomes, because we know that the only outcomes that there can be are orange, purple, brown, and blue. So there’re the only four colours they can land on.

So therefore, the probability of each of these added together must be equal to one. And we can use this to help us solve the problem, cause therefore what we can say is that two 𝑥 plus 𝑥 plus 0.42 plus 0.37 must be equal to one. And that’s because, as we said, the probabilities sum to one.

So now what we have is an equation that we can actually solve to find the value of 𝑥, which is gonna be the probability of brown. So now what we need to do is just combine like terms. So we’ve got two 𝑥 plus 𝑥, which gives us three 𝑥. And then we have 0.42 plus 0.37, which gives us 0.79. So then we’re left with the equation three 𝑥 plus 0.79 is equal to one.

So now the next stage to actually solve the equation is subtract 0.79 from each side. And this gives us three 𝑥 is equal to 0.21, because one minus 0.79 is equal to 0.21. And then if we actually divide each side by three, because we want to find a single 𝑥, we’re gonna get 𝑥 is equal to 0.07.

It’s worth noting though, at this point, this is actually where a common mistake is made, because often students will go, “Alright, 0.21, let’s divide that by three; it’s gonna give us 0.7.” However, if you actually multiplied 0.7 by three, you’d get 2.1, not 0.21.

Okay, great! We found what 𝑥 is. So we know what the probability of brown is. So now what we can do is actually move to the probability of purple and try and calculate what that’s going to be. Well, if we look at the probability of purple — so we get back to the left-hand side — we can say that this was two 𝑥. So therefore, what it’s gonna be is two multiplied by our 𝑥 value. So it’s gonna be two multiplied by 0.07, which is gonna be equal to 0.14.

We can double-check that’s right because it said that the probability of landing on purple was twice the probability of landing on brown. Well, our probability of landing on brown is 0.07, multiply this by two, 0.14. So therefore, what we’ve done is we’ve actually solved the problem, because it says complete the table, given that the probability of landing on purple is twice the probability of landing on brown. And we’ve done that because we’ve got the probably purple is 0.14 and the probability of brown is 0.07.

However, what I want to do at this point is just double-check that we’ve actually got these answers correct. And to actually double-check our answer, what we can do is actually add together all our probabilities. So we have 0.42 plus 0.14, which is gonna give us 0.56, plus 0.07 is gonna give us 0.63, then plus 0.37 gives us one. So yeah, the probabilities do sum to one. So therefore, we can say that, yes, the probability of purple is 0.14 and the probability of brown is 0.07.