Video: Pack 4 • Paper 2 • Question 12

Pack 4 • Paper 2 • Question 12

05:52

Video Transcript

Harry did a survey asking 100 people which colors they liked from blue, green, and red. 38 people like all three colors. 49 people like green and blue. Nine people like blue and red, but not green. 41 people like green and red. 14 people like only blue. 67 people like red. Two people do not like any of these three colors. Harry randomly chooses one of the participants from his survey. Work out the probability that this person likes the color green.

This is a Venn diagram question. It can be hard to spot whether to use a Venn diagram, tree diagram, or some other method to solve a probability question. However, the more of these questions you complete, the more easily you’ll be able to spot the correct technique.

In general, we use a Venn diagram when there are a list of events, some of which overlap. In this case, for example, some people like red, some like green, and some like both, represented by an overlap. Let’s start by drawing three circles to represent the three colors. We enclose them in a rectangle.

This is called the universal set. And it represents everyone who was questioned. In this case, that’s 100 people. We should generally try to follow the information in order, starting at the inside where all three circles overlap. This is called the intersection. We’re told that 38 people like all three colors. So we can put 38 in the middle.

We’re next told that 49 people like green and blue. The people who like green and blue are represented by the intersection of the green circle and the blue circle. Notice that it includes the people who like all three colors. So we need to subtract 38 from 49 to find the number who like green and blue, but not red. 49 minus 38 is 11. So we put 11 on the intersection between the green and the blue circle.

Next, we’re told that nine people like blue and red, but not green. Nine goes on the intersection then between the blue and the red circle. We don’t need to perform any extra calculations for this one since we’re told that these nine people do not like green. They’re not included in the intersection of all three circles.

41 people like green and red. Notice again that this includes people who like all three colors. We need to subtract 38 from 41. That gives us three people who like green and red, but not blue. 14 people like only blue. This means those people do not like the other two colors. So we can place the 14 here.

Next, we’re told that 67 people like red. Looking at our diagram, we can see that that includes all the people who like red and green, red and blue, and all three colors. We, therefore, have to subtract 38, nine, and three from 67. That gives us 17 people who like red, but not blue or green.

The final piece of information we’re given is that two people do not like any of these three colors. This goes on the outside of our circles. We need next to find the number of people who like only green, but not blue or red. To find this number, we’ll subtract all of the values in our Venn diagram from 100 since that was the total number of people questioned. 100 minus 14 plus two plus 11 plus nine plus 38 plus three plus 17 is six. We can, therefore, add the number six to our diagram. There are six people who like green, but not blue or red.

The question is asking us to find the probability a person chosen at random likes the color green. That’s everyone in this circle. Adding together all of the numbers in this circle gives us a total of 58 people who like green. The question is asking us though to find the probability that the person chosen likes the color green. In that case, we write it as 58 out of 100 since there were 100 people asked altogether. 58 out of 100 is equal to 0.58. The probability that a person chosen at random likes the color green is 0.58.

Given that the person Harry chooses likes the color green, find the probability that they also like either blue or red, but not both.

The key phrase in this part of the question is “Given that.” That means we are no longer looking at everyone asked, just a small section. In this case, we’re told that the person likes green. So we narrow our result down to just the people who like green. We already calculated this to be 58 people.

Out of that group of people, we want to find the number who like blue or red, but not both. That’s these two intersections. 11 plus three is 14. So the probability that that person likes blue or red, but not both given that they like the color green is 14 out of 58. By dividing both the numerator and the denominator of this fraction by two, we can simplify our answer to seven twenty-ninths.

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