# How many students are there in the class, what is the probability that a student likes regular rock and the probability that a student likes techno given that they do not like Glam Rock in the...

How many students are there in the class, what is the probability that a student likes regular rock and the probability that a student likes techno given that they do not like Glam Rock in the following case:

A university professor surveys his students and he finds out that about 21 of his students like techno music, 18 like Glam rock, 15 like regular rock, 16 like techno and Glam rock, 10 like Glam rock and regular rock, 11 like techno and regular rock, 8 like all of those three genres, and 3 don't like any of them.

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### 1 Answer

A university professor surveys his students and he finds out that 21 students like techno music, 18 like Glam rock, 15 like regular rock, 16 like techno and Glam rock, 10 like Glam rock and regular rock, 11 like techno and regular rock, 8 like all of those three genres, and 3 don't like any of them.

If the number of students that like Glam rock is set A, `N(A)` = 18. The number of students that like regular rock is `N(B)` = 15 and the number of students that like techno is `N(C)` = 21. `N(AnnC)` = 16, `N(AnnB)` = 10 and `N(BnnC)` = 11 and `N(AnnBnnC)` = 8

The number of students in the class is `N(A)` + `N(B)` + `N(C)` - `N(AnnB)` - `N(AnnC)` - `N(BnnC)` + `N(AnnBnnC)` = 18 + 15 + 21 - 16 - 10 - 11 + 8 = 25

The probability that a student chosen likes regular rock is `15/25` = 0.6

The probability that they like techno given that they do not like Glam Rock is `(21 - 16)/25 = 5/25 = 0.2`

**There are 25 students in the class. The probability that a student likes regular rock is 0.6 and the probability that a person likes techno but not Glam rock is 0.2**