Question

In: Statistics and Probability

suppose given the three pairwise independent events, all three of which cannot simultaneously occur. Assuming that...

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

Solutions

Expert Solution

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.


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