Answer to Question #225968 in Statistics and Probability for Sony

Question

Out of a total of 130 students, 60 are wearing hats to class, 51 are wearing scarves, and 30 are wearing both hats and scarves. Of the 54 students who are wearing sweaters, 26 are wearing hats, 21 are wearing scarves and 12 are wearing both hasts and scarves. Everyone wearing neither a hat nor a scarf, is wearing gloves. a. How many students are wearing gloves? b. How many students not wearing a sweater are wearing hats but not scarves? c. How many students not wearing a sweater are wearing neither a hat nor a scarf? Solve using principle of Inclusion & Exclusion.

Answer 1

a) Everyone wearing neither a hat nor a scarf is wearing gloves.

$n(H \cup S)= n(H)+n(S) - n(H \cap S)$

= 60+51-30

=81

n(H' n S') = 130-81 =49

49 students are wearing gloves.

b)26 students are wearing hat and sweater, hence 60 - 26 = 34 students are wearing hat but not scarves.

12 students wear hat, sweater and scarves together.

So, 30-12=18 students wear sweater and scarves but not sweater.

So, no of students wear hat but not scarves and sweater is 34-18 = 16

c) No of students not wearing sweater =130-54 = 76

No of students wear hat or scarves but not sweater = 81-(26+21-12) = 46

No of students not wearing hat, scarves and sweater =76-46 =30