# Video: Using Venn Diagrams to Describe Sample Spaces

At a restaurant, 34 of the guests ordered a meat main course and 26 ordered a vegetarian main course. 27 people also ordered a dessert; 11 of them had ordered a vegetarian main course. Which of the following Venn diagrams describes the combinations of meals that were eaten? Find the probability of a randomly selected guest having a meat main course and no dessert. Give your answer as a fraction in its simplest form.

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### Video Transcript

At a restaurant, 34 of the guests ordered a meat main course and 26 ordered a vegetarian main course. 27 people also ordered a dessert; 11 of them had ordered a vegetarian main course. Which of the following Venn diagrams describes the combinations of meals that were eaten?

There is also a second part to this question that we will look at later. We are told that 34 of the guests ordered a meat main course. This means that the sum of the numbers in the meat circle must be 34. 18 plus 16 is equal to 34. 14 plus 20 is also equal to 34. 18 plus 20 is equal to 38. Therefore, this Venn diagram cannot be correct. We were also told that 26 people ordered a vegetarian main course. This means that the two numbers inside this circle must sum to 26. 11 plus 15 is equal to 26. However, 11 plus 16 equals 27. So this Venn diagram is also incorrect.

The next piece of information we were told was that 27 people ordered a dessert. Therefore, the two numbers in this circle must sum to 27. 16 plus 11 is equal to 27, whereas 20 plus 11 is equal to 31. The correct Venn diagram is the first one. 34 guests ordered a meat main course, 26 ordered vegetarian, and 27 people ordered a dessert. This Venn diagram also has 11 out of the 27 people who ordered the dessert that also ordered a vegetarian main course. We will now clear some room for the second part of this question.

Find the probability of a randomly selected guest having a meat main course and no dessert. Give your answer as a fraction in its simplest form.

The probability of a random event occurring can be written as a fraction. The numerator is the number of successful outcomes and the denominator is the number of possible outcomes. In this question, the numerator will be those guests that had a meat main course and no dessert. The denominator or possible outcomes will be the total number of guests. We recall that there were 34 guests that ordered a meat main course. Of these, 16 ordered a dessert. This means that the number 18 corresponds to those guests that had a meat main course and no dessert. In total, there were 60 guests.

We can calculate this by adding 34 and 26, the number of people that had a meat main course and those that had a vegetarian main course. Alternatively, we could add the four numbers 18, 16, 11, and 15 from the Venn diagram. The probability is therefore equal to 18 out of 60. Both of these numbers are divisible by six. 18 divided by six is equal to three. And 60 divided by six is equal to 10. As these two numbers have no common factor apart from one, the probability of a guest having a meat main course and no dessert in its simplest form is three out of 10 or three-tenths.