Question

suppose given the three pairwise independent events, all three of which cannot simultaneously occur. Assuming that they all have the same probability x , determine the largest possible value of x.

Answer 1

ANSWER:-

Determine the largest possible value of x:-

• Given 3 occasions a chance to be A, B and C.
• They can't happen at the same time. Along these lines, they are totally unrelated occasions. Along these lines,
• P(ABC) =0
• P(AB) =0
• P(BC) =0
• P(AC) =0
• A, B and C are pairwise free occasions.
• Hence, P(AB) =P(A)P(B),
• Similarly for P(BC) and P(AC)
• The likelihood of every one of the 3 occasions is same and it is 'x'.
• P(A) =x.
• P(B)=x.
• P(C) =x.
• In this way, 'x' lies somewhere in the range of 0 and 1, comprehensive. That is, 'x' has a place with [0, 1].
• Let the biggest esteem 'x' can take is 1. Be that as it may, on the off chance that P(A) =1, at that point, P(AB) =P(A)P(B) P(A)=1,P(B)=0 then P(AB)=0 in light of the fact that P(AB) =0 thus, P(B) must be 0.
• Yet, at that point P(A) P(B). Along these lines, let us take P(A) =1/3. Presently, P(B) must be 0 so that P(AB) =0. Again P(A) P(B). Along these lines, any estimation of 'x' other than 0 prompts logical inconsistency of correspondence of 3 probabilities.
• Presently, let us take
• P(A) =x =0
• P(B) =x =0
• P(C) =x =0
• In this way, 3 conditions are fulfilled just when x =0.

1). P(ABC)=P(AB) =0

P(ABC)=P(BC)=0

P(ABC)=P(AC) =0

2) P(AB) =P(A)P(B)

P(A)=0,P(B)=0 then,

P(AB)=0

and comparably,

P(BC) =0

P(AC) =0

3)

P(A) =x =0

P(B) =x =0

P(C)=x =0

In this way, the biggest and just conceivable estimation of 'x' is 0.