Video: GCSE Mathematics Foundation Tier Pack 1 • Paper 1 • Question 12

GCSE Mathematics Foundation Tier Pack 1 • Paper 1 • Question 12

04:58

Video Transcript

There are two spinners divided into equal sections. Laurence spins both spinners and adds the two numbers together. Calculate the probability that the sum is greater than five.

For this question, we’re going to be dealing with probabilities. And we should first outline the information that we know. Firstly, the question has told us that each spinner has been divided into equal sections. From the diagram, we can see that each spinner has three sections.

To answer this question, we’re going to assume that each spinner will stop in a random position around the circle. From this, we conclude that spinner A is equally likely to land on a two, a four, or a six and spinner B is equally likely to land on a one, a three, or a five.

When dealing with probabilities, it can be useful to observe all of the possible outcomes of the given events. One of the tools that we can use to find these outcomes is a two-way table. Horizontally along our table, we’ve listed the three possible outcomes that can be obtained from spinner A, which are two, four, and six. Vertically along our table, we’ve listed the three possible outcomes that can be obtained from spinner B, which are one, three, and five.

Now the question tells us that Laurence spins both spinners and adds together the two numbers. In order to fill in our two-way table, we can therefore add together the number obtained on spinner A with the number obtained on spinner B. For example, in the top left of our table, if Laurence spins a two on spinner A and a one on spinner B, the total will be two plus one, which is equal to three. If, instead, Laurence spins a four on spinner A and a one on spinner B, the total will be four plus one, which is equal to five.

We can continue to fill in this table with all of the sum totals from the two spins. Once we have done this, we will have obtained a sum total for all of the possible outcomes of the spins. If we look at our table, we can see that the total number of outcomes, which we’ve called 𝑛 total, is nine. We can understand this by observing that both spinner A and spinner B have three possible outcomes. And three times three is equal to nine.

The question now asked us to find the probability that the sum of Laurence’s spins is greater than five. In order to move forward, we should now look at all of our possible outcomes and see which of them satisfies this criteria.

Here we should be careful to note that we are looking for sums where the outcome is greater than but not greater than or equal to five. We should therefore discount a sum of five itself from the outcomes that we say satisfy the criteria.

Now going back to our two-way table, we can see three of our outcomes have a sum of seven, which is indeed greater than five. Two more of our outcomes have a sum of nine, which is also greater than five. And finally, the last outcome which satisfies our criteria has a sum of 11. The remaining outcomes in our table have sums of three or five, and these are either less than or equal to five. We will therefore ignore these outcomes since they do not satisfy the criteria.

Looking at the table, we can now see that there are six outcomes where the sum, which we’ve chosen to call 𝑥, is greater than five. Let us now remember that both of our spinners are equally likely to land on each of their numbers. In other words, the spin is a fair.

From this, we can conclude that each of the nine possible outcomes is also equally likely. This allows us to say the following. The probability that the sum of both spins, which we’ve called 𝑥, is greater than five is equal to the number of outcomes where the sum is greater than five divided by the total number of outcomes. We have just found that there are six outcomes where the sum is greater than five, and there are nine outcomes in total. Substituting these numbers in, we find that our probability is six over nine.

It’s good practice to simplify your fractions. And this fraction can be simplified by dividing both the top and bottom half by three. We have therefore found that the probability that the sum of Laurence’s spins is greater than five is equal to two over three, or two-thirds.

Before we go, a quick check for questions like this is whether your probabilities are less than one. This is because one represents a probability of a 100 percent, which is the maximum value of probabilities of this type. Since our answer of two over three is a fraction where the numerator is less than the denominator, it is indeed less than one. It therefore satisfies the conditions of our check, and we can be more confident in our answer of two over three.

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