Video: AQA GCSE Mathematics Higher Tier Pack 2 • Paper 2 • Question 22

A vegetable stall has boxes of lettuces and cabbages. Box A contains 12 lettuces and 8 cabbages. Box B contains 9 lettuces and 14 cabbages. Frank picks a vegetable at random from box A and places it into box B. Regina then picks a vegetable at random from box B. Show that 𝑃(Frank picks a lettuce) = 𝑃(Regina picks a cabbage).

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Video Transcript

A vegetable stall has boxes of lettuces and cabbages. Box A contains 12 lettuces and eight cabbages. Box B contains nine lettuces and 14 cabbages. Frank picks a vegetable at random from box A and places it into box B. Regina then picks a vegetable at random from box B. Show that the probability that Frank picks a lettuce is equal to the probability that Regina picks a cabbage.

This might look a little tricky. Let’s begin by considering what we actually know. Frank is choosing a vegetable from box A and then placing it into box B, then Regina picks a vegetable from that second box. We can actually represent this information in a tree diagram. Whilst this isn’t the only way of solving the problem, it’s certainly the best way to ensure that we don’t lose any information as we go through it.

The first event is Frank’s choice of vegetable from box A. He could choose a lettuce or a cabbage. We can find the total number of vegetables in the box by adding together the number of lettuces and the number of cabbages. 12 plus eight is 20, so there’s 20 pieces of vegetable in the box. 12 of these are lettuces, so the probability that Frank chooses a lettuce is 12 over 20. Eight of them are cabbages, and the probability that he chooses a cabbage is eight over 20.

The second event is Regina’s choice of vegetable from box B. No matter the outcome from the first event, she could either choose a lettuce or a cabbage. But we’re going to need to think really carefully about the probabilities of her doing so. Let’s begin by using this first branch. We have assumed that Frank has chosen a lettuce. That means in box B there is now one more lettuce. There are now 10 lettuces in box B, but still 14 cabbages. That means there’s a total of 24 vegetables in that box, and the probability that Regina chooses a lettuce is 10 over 24. There are 14 cabbages, so the probability she chooses a cabbage is 14 over 24.

Let’s now follow the second branch for Frank’s outcome. In that branch, he’s chosen a cabbage and then put it into box B. There is now one more cabbage in that box, meaning that that box has nine lettuces and 15 cabbages. That is still a total of 24 pieces of vegetable. So the probability that Regina chooses a lettuce now is nine over 24, and the probability she chooses a cabbage is 15 over 24. We’re going to use our tree diagram to find the probability that Regina picks a cabbage. There are two ways for her to do so: either Frank picks a lettuce and Regina chooses a cabbage or Frank chooses a cabbage and Regina chooses a cabbage.

To find the probability that Frank chooses a lettuce and Regina chooses a cabbage, let’s recall the and rule. This says for two independent events A and B, the probability of them both occurring is the probability of A multiplied by the probability of B. This means the probability that Frank chooses a lettuce and Regina chooses a cabbage is 12 over 20 multiplied by 14 over 24. We can cross cancel by dividing by 12 and 24 by two, and we can divide both 14 and 20 by two. That gives us seven over 20. We multiply the numerators together and then we multiply the denominators.

Next, we want to find the probability that they both choose a cabbage. That’s eight over 20 multiplied by 15 over 24. We can simplify by dividing both eight and 24 by eight, and we can divide both 15 and 20 by five. Three over three simplifies to one, so the probability of choosing a cabbage and a cabbage is one-quarter. The next rule says that for two events that are mutually exclusive — that is, they can’t happen at the same time — the probability of A or B occurring is found by adding their probabilities. That means we can find the probability that Frank chooses a lettuce and Regina chooses a cabbage or the probability that Frank chooses a cabbage and Regina chooses a cabbage by adding seven twentieths and one- quarter.

Now you might have noticed that have we not cross cancelled earlier, we would have been able to add these fractions fairly easily. As it is though, we need to now make the denominators the same. We’re going to do this by multiplying the numerator and the denominator of the second fraction by five. That gives us five twentieths. Seven twentieths plus five twentieths is twelve twentieths. We were trying to choose [show] that the probability that Frank picks a lettuce is equal to the probability that Regina picks a cabbage. We already worked out that the probability that Frank picks a lettuce is twelve twentieths, and we’ve just shown that the probability that Regina picks a cabbage is also twelve twentieths. So we are done. The probability that Frank picks a lettuce is equal to the probability that Regina picks a cabbage, which is equal to twelve twentieths.

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