In a survey of a group of people, it was found that 65% of them liked comedy movies, 63% liked action movies, 33% liked both types of movies and 150 people did not like both types of movies.

1. Draw a Venn-diagram to illustrate the above information.
2. Find the number of people participated in the survey.
3. Find the number of people who liked only one type of movies.

Answer: (i) - , (ii) 3000 and (iii) 1860

Solution:

Let U be the set of total people surveyed.

Let C and A be the sets of people who liked comedy movies and action movies, respectively.

Given,

$n(U) = 100%$

$n(C) = 65%$

$n(A) = 63%$

$n(C \cap A) = 33%$

$n(\overline{C \cup A}) = 150$

Using formula,

$n(U) = n(C) + n(A) - n(C \cap A) + n(\overline{C \cup A})$

$or, 100% = 65% + 63% - 33% + n(\overline{C \cup A})$

$or, n(\overline{C \cup A}) = 100% - 95%

$\therefore n(\overline{C \cup A}) = 5%$

We know,

$n(\overline{C \cup A}) = 150$

$or, 5% of n(U) = 150$

$or, \frac{5}{100} × n(U) = 150$

$or, n(U) = \frac{150×100}{5}$

$\therefore n(U) = 3000$

Now,

$n_o(A) + n_o(C) = n(A) + n(B) - 2 n(C \cap A)$

$= 65% + 63% - 2×33$

$= 65% + 63% - 66%$

$= 62%$

$= 62% of n(U)$

$= \frac{62}{100} × 3000$

$= 62 × 30$

$= 1920$

Hence, the required total number of people surveyed was 3000 and 1920 people of them liked only one type of movies.

Explanation:

People who liked only one type of movies is the sum of people who liked either only Comedian Movies or Action movies. So, we added $n_o(C)$ and $n_o(A)$.

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