Given: Sample space is the set of first 500 natural numbers.
n (S) = 500
Let ‘A’ be the event of choosing the number such that it is divisible by 3
n (A) = [500/3]
= [166.67]
= 166 {where [.] represents Greatest integer function}
P (A) = n (A) / n (S)
= 166/500
= 83/250
Let ‘B’ be the event of choosing the number such that it is divisible by 5
n (B) = [500/5]
= [100]
= 100 {where [.] represents Greatest integer function}
P (B) = n (B) / n (S)
= 100/500
= 1/5
Now, we need to find the P (such that number chosen is divisible by 3 or 5)
P (A or B) = P (A ∪ B)
By using the definition of P (E or F) under axiomatic approach (also called addition theorem) we know that:
P (E ∪ F) = P (E) + P (F) – P (E ∩ F)
∴ P (A ∪ B) = P (A) + P (B) – P (A ∩ B)
[Since, we don’t have value of P(A ∩ B) which represents event of choosing a number such that it is divisible by both 3 and 5 or we can say that it is divisible by 15.]
n(A ∩ B) = [500/15]
= [33.34]
= 33
P (A ∩ B) = n(A ∩ B) / n (S)
= 33/500
∴ P (A ∪ B) = P (A) + P (B) – P (A ∩ B)
= 83/250 + 1/5 – 33/500
= [166 + 100 – 33]/500
= 233/500
