A survey of 121 college students was taken to determine the musical styles they
ID: 3194982 • Letter: A
Question
A survey of 121 college students was taken to determine the musical styles they liked. Of those, 35 students listened to rock, 32 to classical, and 26 to jazz. Also, 7 students listened to rock and jazz, 16 to rock and classical, and 7 to classical and jazz. Finally, 5 students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region. Use the completed Venn Diagram to answer the following questions.
How many listened to only rock?
How many listened to classical and jazz, but not rock?
How many listened to classical or jazz, but not rock?
How many listened to only one style of music?
How many listened in exactly two styles of music?
How many did not listen to any music styles?
Explanation / Answer
P(A U B U C) = P(A) + P(B) + P(C) - P(A^B) - P(B^C) - P(C^A) + P(A^B^C)
^ is intersection.
P(A U B U C) = P(A U B) + P(C) - P((A U B)^C)
= P(A) + P(B) - P(A^B) + P(C) - P((A^C) U (B^C))
= P(A) + P(B) - P(A^B) + P(C) - [P(A^C) + P(B^C) - P((A^C)^(B^C))]
= P(A) + P(B) + P(C) - P(A^B) - P(A^C) - P(B^C) + P(A^B^C)
You may have noticed that you find the probability by adding the probabilities of the individual events, then taking away the probabilities of each combination of two events, and finally adding the probability for all three to happen. This pattern is called the inclusion-exclusion principle, and it applies to the union of any number of probabilities.
if we consider a is rock ,b is a classical ,c is a jazz then
p(A)=35
P(B)=32
P(C)=26
P(A^B)=7
P(B^C)=16
P(C^A)=7
P (A^B^C)=5
i)the probapility lesion only rock p (rock)=35
ii)the probablity leasion both classic and jazz but not rock =p(classic and jazz)=7
iii)the probablity leasion both classic or jazz =p(classic or jazz)=
p(B U C)=P(B)+P(C)-P (B AND C)=
=32+26-7=51
IV)HOW MANY LISTENED ALL TYPE OF MUSIC=
P(A U B U C) = P(A U B) + P(C) - P((A U B)^C)
= P(A) + P(B) - P(A^B) + P(C) - P((A^C) U (B^C))
= P(A) + P(B) - P(A^B) + P(C) - [P(A^C) + P(B^C) - P((A^C)^(B^C))]
= P(A) + P(B) + P(C) - P(A^B) - P(A^C) - P(B^C) + P(A^B^C)
=35+32+26-7-16-7+5
=68
HOW many listened only one style of music=121-68=53
v)how many listened in exactly two styles of music
=7+16+7
=30
vi)how many did not listen to any music styles
=
P(A U B U C) =
= P(A) + P(B) + P(C) - P(A^B) - P(A^C) - P(B^C) + P(A^B^C)
=35+32+26-7-16-7+5
=68
how many listened only one style of music=100-68=32
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