Three students S₁, S₂ and S₁ are given a problem to solve. Consider the following events : U: At least one of S₁, S₂ and S₃ can solve the problem, V: S₁ can solve the problem, given that neither S ₂ nor S ₃ can solve the problem, W: S₂ can solve the problem and S₃ cannot solve the problem, T: S₃ can solve the problem. For any event E , let P(E) denote the probability of E . If P(U)= 1 2 , P(V)= 1 10 and P(W)= 1 12 ,

Step-by-step solution

aligned & P ( U )=1- P ( ~S ₁^ S ₂^ S ₃^ )= 1 2 & P ( ~S ₁^ S ₂^ S ₃^ )= 1 2 ; P ( ~S ₁^ ) P ( ~S ₂^ ) P ( ~S ₃^ )= 1 2 & (1- P ( ~S ₁ ) ) (1- P ( ~S ₂ ) ) (1- P ( ~S ₃ ) )= 1 2 (1) aligned aligned & P ( V )= P ( S ₁ ~S ₂^ S ₃^ ) P ( S ₂^ S ₃^ ) = 1 10 & P ( S ₁ ) P ( S ₂^ ) P ( S ₃^ )= 1 10 P ( S ₂^ ) P ( S ₃^ ) & P ( S ₁ )= 1 10 aligned aligned & P ( ~W )= P ( ~S ₂ ~S ₃^ )= 1 12 & P ( ~S ₂ ) P ( ~S ₃^ )= 1 12 & P ( ~S ₂ ) (1- P ( ~S ₃ ) )= 1 12 ...(2) aligned Eq. (1) (1- 1 10 ) (1- P ( ~S ₂ ) ) (1- P ( ~S ₃ ) )=