For the three events A, B and C, P(exactly one of the events A or B o

1.	For the three events A, B and C, P(exactly one of the events A or B occurs) = P(exactly one of B or C occurs = p(exactly one of C or A occurs) = P (all the three events occur simultaneously) = p2 where 0 < p < \(\frac{1}{2}\) . Then, find the probability of occurrence of at least one of the three events A, B and C.
Answer» Given, P (exactly one of A or B occurs) = p P (Exactly one of B or C occurs) = p_' P (exactly one of C or A occurs) = p_' and p (All three occurs simultaneously) = p². i. e., P(A) + P(B) – 2P(A ∩ B) = p…(i) P(B) + P(A) – 2P(B ∩ C) = p…(ii) P(C) + P(A) – 2P(C ∩ A) = p …(iii) and, P(A ∩ B ∩ C) = p² …(iv) (i) + (ii) + (iii) ⇒ P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(C ∩ A) = \(\frac{3}{2}p\) We know, P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(A ∩ C) + P(A ∩ B ∩ C) \(=\frac{3}{2}p + p^2\) \(= \frac{3p + 2p^2}{2}\) ∴ The probability that at least one of the three events A, B and C occurs is \(\frac{3p + 2p^2}{2}\).

For the three events A, B and C, P(exactly one of the events A or B occurs) = P(exactly one of B or C occurs = p(exactly one of C or A occurs) = P (all the three events occur simultaneously) = p2 where 0 < p < \(\frac{1}{2}\) . Then, find the probability of occurrence of at least one of the three events A, B and C.

Answer»

Given,

P (exactly one of A or B occurs) = p

P (Exactly one of B or C occurs) = p_'

P (exactly one of C or A occurs) = p_'

and p (All three occurs simultaneously) = p².

i. e., P(A) + P(B) – 2P(A ∩ B) = p…(i)

P(B) + P(A) – 2P(B ∩ C) = p…(ii)

P(C) + P(A) – 2P(C ∩ A) = p …(iii)

and, P(A ∩ B ∩ C) = p² …(iv)

(i) + (ii) + (iii)

⇒ P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(C ∩ A) = \(\frac{3}{2}p\)

We know,

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(A ∩ C) + P(A ∩ B ∩ C)

\(=\frac{3}{2}p + p^2\)

\(= \frac{3p + 2p^2}{2}\)

∴ The probability that at least one of the three events A, B and C occurs is \(\frac{3p + 2p^2}{2}\).