Video: AQA GCSE Mathematics Higher Tier Pack 5 • Paper 1 • Question 20

There are 13 marbles in a bag. 3 of the marbles are red and the rest are black. Sasha randomly takes out two marbles from the bag without replacement. Work out the probability that the two marbles are the same colour.

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Video Transcript

There are 13 marbles in a bag. Three of the marbles are red and the rest are black. Sasha randomly takes out two marbles from the bag without replacement. Work out the probability that the two marbles are the same colour.

The word without is written in bold type, which means it must be pretty important. If Sasha takes the marbles from the bag without replacement, it means she doesn’t put the first marble back in the bag before she selects the second. This means that there are only 12 marbles available for her second choice. That’s 13 minus one because she’s already taken one marble out of the bag and not put it back. It’s really important that we remember this as we go through the question.

We can use a tree diagram to help us answer this question. We’ll start off with Sasha’s first marble, which can be either red or black. Then, we have the second marble, which can also be either red or black. Now let’s think about the probabilities for each of these branches on the tree diagram.

To start off with, there are 13 marbles in the bag and three of them are red. Sasha chooses her marbles at random, which means the probability of her selecting each marble is the same. So the probability that she chooses a red marble is three out of 13. We’re told that the rest of the marbles are black. So this means that there are 10 black marbles in the bag at the start. That’s 13 minus three. So the probability that the first marble Sasha chooses is black is 10 out of 13.

What about the branches on the second stage of the tree diagram? Remember Sasha doesn’t put the first marble back in the bag, which means that there are now only 12 marbles available to be chosen. So the denominators for the fractions on the second set of branches are all going to be 12. If the first marble that Sasha took was red, then there’re only two red marbles left in the bag. So the probability that the second marble will be red if the first was red is two twelfths, two out of 12. But we haven’t taken any black marbles out of the bag in this case. So there were still the 10 black marbles that we started off with, which means the probability that Sasha’s second marble is black if the first marble was red is 10 out of 12.

Finally, let’s consider what happens if the first marble that Sasha took was black. There are still the same number of red marbles in the bag as there were to begin with. So the probability that Sasha’s second marble is red if the first marble was black is three out of 12. But this time, there are only nine black marbles left because Sasha took one of them out of the bag. So the probability that her second marble is black if the first marble was black is nine out of 12. Notice that the probabilities on every set of branches on this tree diagram sum to one.

Now the question asked us to work out the probability that the two marbles Sasha chooses are the same colour. There are two ways that Sasha could do this. She could either get two red marbles or she could get two black marbles. To find the probabilities at the end of the branches on a tree diagram, we multiply together the probabilities along the route that we need to take to get there.

So to find the probability that the first marble is red and the second marble is red, we multiply three over 13 — that’s the probability the first marble is red — by two over 12 — that’s the probability the second marble is red if the first marble was red. Now let’s simplify this calculation slightly. We can cross cancel a factor of three from the three in the numerator of the first fraction and the 12 in the denominator of the second, giving one and four. We can also cancel within the second fraction. Two and four can both be divided by two, giving one in the numerator and two in the denominator.

To multiply the two fractions together, we multiply the numerators and then multiply the denominators. One multiplied by one is one and 13 multiplied by two is 26. So the probability that both marbles that Sasha chooses are red is one over 26.

To find the probability that both marbles are black, we multiply the probabilities along these branches of the tree diagram. That’s 10 over 13 multiplied by nine over 12. We can simplify the second fraction by dividing both the numerator and denominator by three, giving three over four. And we can cross cancel a factor of two from the 10 in the numerator of the first fraction and the four in the denominator of the second, leaving five over 13 multiplied by three over two. Five multiplied by three is 15 and 13 multiplied by two is 26. So the probability that the two marbles Sasha chooses are both black is 15 over 26.

Finally, we need to combine these two probabilities together. And to do so, we use the OR rule of probability, which tells us that if two events 𝐴 and 𝐵 are mutually exclusive, meaning they can’t happen at the same time, then to find the probability of one event or the other happening, we add their individual probabilities together. So to find the probability that Sasha gets two red marbles or two black marbles, we add one over 26 and 15 over 26.

To add fractions, we need a common denominator which we already have. So then, we just add the two numerators together, giving 16 over 26. However, we can simplify this fraction by dividing both the numerator and denominator by two, giving eight over 13. This fraction can’t be simplified any further. And there’s no need to try and convert it to a decimal.

So we found that the probability that the two marbles are the same colour is eight over 13.

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