Question

A campus radio station surveyed 500 students to determine the types of music they like. The survey revealed that 206 like rock, 161 like country, and 118 like jazz. Moreover, 30 like rock and country, 27 like rock and jazz, 22 like country and jazz, and 11 like all three types of music. What is the probability that a randomly selected student likes neither rock nor country? Note: A Venn diagram may be useful here.

Answer 1

Answer:

163/500 = 0.326

Step-by-step Explanation:

Drawing a Venn diagram for this, we will have 3 circles.  One will represent rock, one will represent country, and one will represent jazz.

There are 11 students that like all 3 types of music.  This means the number 11 goes in the intersection of all 3 circles.

There are a total of 30 students that like rock and country; taking out the 11 that like all 3, this leaves 30-11 = 19 students in the intersection of just rock and country.

There are a total of 27 students that like rock and jazz; taking out the 11 that like all 3, this leaves 27-11 = 16 students in the intersection of just rock and jazz.

There are a total of 22 students that like country and jazz; taking out the 11 that like all 3, this leaves 22-11 = 11 students in the intersection of just country and jazz.

There are 206 students that like rock; taking out the 11 that like all 3, the 19 that like rock and country, and the 16 that like rock and jazz, we have

206-(11+19+16) = 206-(46) = 160 in just rock.

There are 161 students that like country; taking out the 11 that like all 3, the 19 that like rock and country, and the 11 that like country and jazz, we have

161-(11+19+11) = 161-(41) = 120 in just country.

There are 118 students that like jazz; taking out the 11 that like all 3, the 16 that like rock and jazz, and the 11 that like country and jazz, we have

118-(11+16+11) = 118-38 = 80 in just jazz.

This leaves

500-(80+120+160+11+11+19+16) = 500-417 = 83 students that like none of the 3 types of music.

This means for the probability that a student likes neither rock nor country, they either like just jazz or none of the 3; this is (83+80)/500 = 163/500 = 0.326.

Answer 2

The probability that a randomly selected student likes neither rock nor country is;

P(R' ∩ C') = 0.326

Let us denote them as follows;

Number that like rock music be R

Number that like Country music be C

Number that like Jazz music be J

Thus, as we are given we have;

R = 206

C = 161

J = 118

R ∩ C = 30 - 11 = 19

R ∩ J = 27 - 11 = 16

C ∩ J = 22 - 11 = 11

R ∩ C ∩ J = 11

• Thus, number that liked only jazz music is;

n(only jazz) = J - [(R ∩ J) + (C ∩ J) + (R ∩ C ∩ J)]

n(only jazz) = 118 - (16 + 11 + 11)

n(only J) = 80

• Number of students that liked only Rock music;

n(only Rock) = R - [(R ∩ C) + (R ∩ J) + (R ∩ C ∩ J)]

n(only Rock) = 206 - (19 + 16 + 11)

n(only R) = 160

• Number of students that liked only country music;

n(only country) = C - [(R ∩ C) + (C ∩ J) + (R ∩ C ∩ J)]

n(only country) = 161 - (19 + 11 + 11)

n(only C) = 120

Now, the number of students that like neither of the 3 music is;

A' ∩ B' ∩ C' =  500 - [(R ∩ C) + (R ∩ J) +  (C ∩ J) + (R ∩ C ∩ J) + n(only J) + n(only R) + n(only C)]

⇒ 500 - (80 + 120 + 160 + 19 + 16 + 11 + 11)

⇒ A' ∩ B' ∩ C' = 83

Thus, number of students that likes neither rock nor country is;

n(A' ∩ B') = (A' ∩ B' ∩ C') + n(only C)

n(A' ∩ B') = 83 + 80

n(A' ∩ B') = 163

Thus, probability that a randomly selected student likes neither rock nor country = 163/500 = 0.326

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