UPSC NDA Math Model Paper 7

Given: E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5}
Here we have to find the value of P ((E ∪ F) | G)
As we know that, P ((A ∪ B) | F) = P (A | F) + P (B | F) - P ((A ∩ B) | F)
⇒ P ((E ∪ F) | G) = P(E | G) + P (F | G) - P ((E ∩ F) | G) .........(1)
As we know that, P‌(A|‌B)=

P‌(A‌∩B)

P(B)

⇒ P (E | G) = P (E ∩ G) / P (G) = 1/2.........(2)
⇒ P (F | G) = P (F ∩ G) / P (G) = 1/2 ..........(3)
⇒ P ((E ∩ F) | G) = P (E ∩ F ∩ G) / P (G) = 1/4 ........(4)
By substituting (2), (3) and (4) in equation (1), we get
⇒ P ((E ∪ F) | G) = 1/2 + 1/2 - 1/4 = 3/4