Question Video: Using a Venn Diagram to Determine the Probability of the Complement of an Event | Nagwa Question Video: Using a Venn Diagram to Determine the Probability of the Complement of an Event | Nagwa

# Question Video: Using a Venn Diagram to Determine the Probability of the Complement of an Event Mathematics • Third Year of Preparatory School

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The students in a music class either play piano, guitar or both, while some play neither piano nor guitar. Let 𝐴 denote the students who play piano and let 𝐵 denote the students who play guitar. Use the given diagram to calculate the probability that a student does not play guitar.

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### Video Transcript

The students in a music class either play piano, guitar or both, while some play neither piano nor guitar. Let 𝐴 denote the students who play piano and let 𝐵 denote the students who play guitar. Use the given diagram to calculate the probability that a student does not play guitar.

On the Venn diagram given, the right-hand circle represents those students who play guitar. There are 15 students that play just guitar and there are two students that play the guitar and the piano. This is represented by the intersection of the two circles and gives us a total of 17 students that play guitar. There are a total of 23 students: the 17 that play guitar, four that play just the piano, and two that play neither instrument. The probability that a student plays the guitar is therefore equal to 17 out of 23.

We are asked to calculate the probability that a student does not play the guitar. One way of doing this would be to simply add the numbers outside of circle 𝐵. In this case, four plus two is equal to six. This tells us that six students do not play the guitar.

An alternative method would be to use our knowledge of the complement of events. The probability of the complement denoted 𝐴 bar or 𝐴 prime is equal to one minus the probability of 𝐴. In our question, the probability of 𝐵 bar not playing the guitar is equal to one minus 17 out of 23. This is equal to six out of 23. The probability that a student in the class does not play the guitar is six out of 23.

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