Video Transcript
In a music class, students learn to play the piano, the guitar, and the drums. Some students play two instruments, some play all three instruments, and some play none of the instruments. Let 𝑃 denote those who play the piano, 𝐺 those who play the guitar, and 𝐷 those who play the drums. Using the given diagram, calculate the probability that a student does not play the piano.
In this question, we are given a Venn diagram. This shows the number of students in a class that play the piano, the guitar, the drums, or a combination of all three. There are also some students who play none of the instruments. In this question, we are asked to calculate the probability that a given student does not play the piano. We will begin by considering those students that do play the piano. There are two students that play just the piano, two that play the piano and guitar, three that play the piano and the drums, and two students that play all three instruments. This gives us a total of nine students that play the piano.
If we add all the numbers in the Venn diagram, we have a total of 31. There are therefore 31 students in the class. The probability of selecting a student who does play the piano is therefore equal to nine out of 31. We could find the number of students who do not play the piano by adding the other numbers in the Venn diagram: 14, five, one, and two. However, an alternative method would be to consider the complement of an event. The complement of event 𝐴 is denoted 𝐴 bar or 𝐴 prime. And the probability of this is equal to one minus the probability of 𝐴. This means that the probability of an event not occurring is one minus the probability that it does occur.
In this question, the probability that a student does not play the piano is one minus nine out of 31. This is equal to 22 out of 31. The numerator 22 is also equal to the sum of the numbers outside of the circle representing piano on the Venn diagram.
