Unit -1 Part 2

Unit -1 Part 2

December 09, 2024

Question 5

a) A survey was carried out among 900 people of a community:

525 people read Madhupark.
450 people read Yabamanch.
75 people didn’t read either of the newspapers.

Subquestions:

i) Show the information in a Venn diagram.
We denote the sets:
$M$ : People who read Madhupark.
$Y$ : People who read Yabamanch.
The total number of people in the universal set $U$ is $900$ .
$n(M \cup Y) = n(U) - n(\text{neither}) = 900 - 75 = 825.$
Let $x$ be the number of people who read both newspapers.
$n(M \cup Y) = n(M) + n(Y) - n(M \cap Y).$
Substitute the values:
$825 = 525 + 450 - x \quad \implies \quad x = 150.$
ii) Find the number of people who read both newspapers.
$n(M \cap Y) = 150.$
iii) Find the number of people who read only one newspaper.
People who read only Madhupark:
$n(M \setminus Y) = n(M) - n(M \cap Y) = 525 - 150 = 375.$
People who read only Yabamanch:
$n(Y \setminus M) = n(Y) - n(M \cap Y) = 450 - 150 = 300.$
Total number of people who read only one newspaper:
$n(\text{only one}) = n(M \setminus Y) + n(Y \setminus M) = 375 + 300 = 675.$

b) According to a survey among 150 people:

90 people like modern songs.
70 people like folk songs.
30 people do not like either of the songs.

Subquestions:

i) Show the information in a Venn diagram.
We denote the sets:
$M$ : People who like modern songs.
$F$ : People who like folk songs.
The total number of people in $U$ is $150$ .
$n(M \cup F) = n(U) - n(\text{neither}) = 150 - 30 = 120.$
Let $x$ be the number of people who like both types of songs.
$n(M \cup F) = n(M) + n(F) - n(M \cap F).$
Substitute the values:
$120 = 90 + 70 - x \quad \implies \quad x = 40.$
ii) Find the number of people who like both songs.
$n(M \cap F) = 40.$
iii) Find the number of people who like only modern songs.
$n(M \setminus F) = n(M) - n(M \cap F) = 90 - 40 = 50.$

c) According to a survey among 360 players:

210 players like volleyball.
180 players like football.
30 players like neither of the games.

Subquestions:

i) Show the information in a Venn diagram.
We denote the sets:
$V$ : Players who like volleyball.
$F$ : Players who like football.
The total number of players in $U$ is $360$ .
$n(V \cup F) = n(U) - n(\text{neither}) = 360 - 30 = 330.$
Let $x$ be the number of players who like both games.
$n(V \cup F) = n(V) + n(F) - n(V \cap F).$
Substitute the values:
$330 = 210 + 180 - x \quad \implies \quad x = 60.$
ii) Find the number of players who like both games.
$n(V \cap F) = 60.$
iii) Find the number of people who like only one game.
Players who like only volleyball:
$n(V \setminus F) = n(V) - n(V \cap F) = 210 - 60 = 150.$
Players who like only football:
$n(F \setminus V) = n(F) - n(V \cap F) = 180 - 60 = 120.$
Total number of players who like only one game:
$n(\text{only one}) = n(V \setminus F) + n(F \setminus V) = 150 + 120 = 270.$

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