Of the 300 grade 10 students of a certain high school, 120 joined Mathematics club, 115 joined Science club and 100 joined English club. Furthermore, 55 joined Mathematics and joined Mathematics and Science club, 50 joined Mathematics and English club and 40 joined Science and English club. Finally, 80 students did not join any of these clubs. If a student is selected from this group, find the probability that the chosen student joined
a) all the three clubs.
b) Mathematics or the English club.
c) any of the three clubs.
d) Science club only.
e) Mathematics and Science club but not English club.
Answer:
a.
n (M ∪ S ∪ E) = n (M) + n (S) + n (E) – n (M ∩ S) – n (M ∩ E) – n (S ∩ E) + n (M ∩ E ∩ S) + 80
300 = 120 + 115 + 100 – 55 – 50 -40 + 80 + n (M ∩ E ∩ S)
n (M ∩ E ∩ S) = 300 – 270
n (M ∩ E ∩ S) = 30
P (M ∩ E ∩ S) = n (M ∩ E ∩ S) / n (X)
P (M ∩ E ∩ S) = 30 / 300
P (M ∩ E ∩ S) = 1 / 10
b.
P (M ∪ E) = n (M) + n (E) / n (X)
P (M ∪ E) = 15 + 10 / 300
P (M ∪ E) =25 / 300
P (M ∪ E) = 1 / 12
c.
P (M ∪ S ∪ E) = n (M) + n (S) + n (E) / n (X)
P (M ∪ S ∪ E) = 15 + 20 + 10 / 300
P (M ∪ S ∪ E) = 45 / 300
P (M ∪ S ∪ E) = 3 / 20
d.
P (S) = n (S) / n (X)
P (S) = 20 / 300
P (S) = 1 / 15
e.
P (M ∪ S) = N (M ∪ S) / n (X)
P (M ∪ S) = 15 + 20 + 55 / 300
P (M ∪ S) = 90 / 300
P (M ∪ S) = 3 / 10
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