Video: Using a Venn Diagram to Determine an Intersection of Events

271 students voted for the types of music they wanted at the school dance. The results are shown in the Venn diagram. Find the probability that a randomly selected student voted for rock and not jazz.

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Video Transcript

271 students voted for the types of music they wanted at the school dance. The results are shown in the Venn diagram. Find the probability that a randomly selected student voted for rock and not jazz.

We know that the probability of an event occurring can be written as a fraction, where the numerator is the number of successful outcomes and the denominator, the number of possible outcomes. In this question, as there were 271 students, our denominator will be 271. We want to find the students that voted for rock and not jazz.

Let’s begin by considering all the students that voted for rock. There are four numbers inside this circle: 70, seven, nine, and six. These four numbers sum to 92. Therefore, there were 92 students who voted for rock. The number 70 corresponds to those students who voted for rock only. The seven represents students that voted for rock and country. The six represents students that voted for rock and jazz, as the section intersects the circles for rock and jazz.

Finally, the nine represents students who voted for all three types of music: rock, country, and jazz. The two sections that represent students who voted for rock and not jazz are therefore the 70 and the seven. 70 plus seven is equal to 77. We can therefore conclude that the probability that a randomly selected student voted for rock and not jazz is 77 out of 271.

If we let 𝑅 be the event that a student voted for rock and 𝐽 be the event that a student voted for jazz, then the probability that a student voted for rock and not jazz can be written as shown. The bar above the 𝐽 represents the complement and stands for those students who did not choose jazz. The n symbol represents the intersection. In this case, we want the students that voted for rock and not jazz.

We could have used one of our probability formulas to calculate this. The probability of 𝐴 and the complement of 𝐵 is equal to the probability of 𝐴 minus the probability of the intersection of 𝐴 and 𝐵. In this question, the probability of rock and not jazz is equal to the probability of rock minus the probability of rock and jazz. From our Venn diagram, we saw that the probability of a student voting for rock was 92 over 271.

The probability of a student voting for rock and jazz includes the nine and the six from our Venn diagram. This means that the probability of a student voting for rock and jazz is 15 out of 271. The denominators of these two fractions are equal, so we simply subtract the numerators. And 92 minus 15 is equal to 77. This confirms that the probability that a student voted for rock and not jazz is 77 out of 271.

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