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.

plz give the steps thx

Answer 1

Let P =Probability

Let 3 events be A, B and C.

They cannot occur simultaneously. So, they are mutually exclusive events. Thus, P(ABC) =P(AB) =P(BC) =P(AC) =0

A, B and C are pairwise independent events.

Thus, P(AB) =P(A)*P(B), P(BC) =P(B)*P(C) and P(AC) =P(A)*P(C)

The probability of all 3 events is same and it is 'x'.

P(A) =P(B) =P(C) =x.

So, 'x' lies between 0 and 1, inclusive. That is, 'x' belongs to [0, 1]. Let the largest value 'x' can take is 1. But if P(A) =1, then, P(AB) =P(A)*P(B) =1*0 =0 because P(AB) =0 and so, P(B) must be 0. But then P(A) P(B). So, let us take P(A) =1/3. Now, P(B) must be 0 so that P(AB) =P(A)*P(B) =1/3*0 =0. Again P(A) P(B). So, any value of 'x' other than 0 leads to contradiction of equality of 3 probabilities.

Now, let us take P(A) =P(B) =P(C) =x =0

Thus, 3 conditions are satisfied only when x =0.

1) P(ABC) =P(AB) =P(BC) =P(AC) =0

2) P(AB) =P(A)*P(B) =0*0 =0 and similarly, P(BC) =P(AC) =0

3) P(A) =P(B) =P(C) =x =0

Therefore, the largest and only possible value of 'x' is 0.