Question

In a high school literature club, there are 3 groups: one is Novel, one in Poetry, and one in Comics. These sections are open to any of the 100 students in the school. There are 25 students in the Novel group, 31 in the Poetry group, and 19 in the Comics group. There are 18 students that are in both Novel and Poetry, 7 that are in both Novel and Comics, and 14 are in both Poetry and Comics. In addition, there are 5 students taking all 3 groups. If a student chosen at random,

a) the probability that he is not included in any of these groups is

b) the probability that he is playing exactly one literature group is

c) When two people are chosen randomly, the probability that at least 1 is included in a group is

Answer 1

a) We have

,

Hence,

Number of students enrolled in at least one group = 41

Number of students enrolled in no group = 100-41 = 59

Hence, if a student is chosen at random, probability that he is not in any of these groups = 59/100 = 0.59

b) no. of students playing only Novel = 25 - 18 - 7 + 5 = 5

no. of students playing only poetry = 31 - 18 - 14 + 5 = 4

no. of students playing only Comics = 19 - 7 - 14 + 5 = 3

hence, no. of playing exactly one literature group = 5+ 4+ 3 = 12

If a student chosen at random, the probability that he is playing exactly one literature group is = 12/100 = 0.12

c) Probability of at least being included in the group = 2 x Probability that one is included and the other is excluded (2 is multiplied because it could be wither of them) + probability that both are included

= 2 x (41/100) x (59/100) + (41/100) x (41/100) = 0.4838 + 0.1681 = 0.6519