If `A, B`, and `C` are independent events such that `P(A)=P(B)=P(C)=p` , then find the probability of occurrence of at least two of `A, B`, and `C`.

Explanation

Let `A`, `B`, `C` denotes occurrence of events


Let `A’,B’` and `C’ `denotes not occurrence.


Given, `P(A) = P(B) = P(C) = p`


`P(A’) = P(B’) = P(C’) =1- p`


`P` (At least two of `A, B, C` occur) `= P(A ∩ B ∩ C’) + P(A ∩ B’ ∩ C) + P(A’ ∩ B ∩ C)`


`therefore` Events are independent


`P` (At least two of A, B, C occur) `= P(A)P(B)P(C’) + P(A)P(B’)P(C)+P(A’)P(B)P(C)`


`P` (At least two of A, B, C occur) `= p*p*(1-p)+p*(1-p)*p+(1-p)*p*p`


...