Respuesta :
We have proven that no two events a and b can be independent unless either a or b is the whole sample space or the empty set below.
A sample space can be defined as the set of all possible outcomes of any experiment.
We are already aware that the outcomes of this die are going to be (1,2,3,..p). If we have a pair of proper events A, B which are independent, this would mean that = A∩B={∅} (i)
This is a contradiction. So, now we suppose that = A∩B=C, for some proper event C (ii)
Using the values of (i) and (ii), we get -
= |C|p=P(A∩B)=P(A)P(B)=|A||B|p2.
= p|C|=|A||B|.
From this, we understand that neither A nor B are going to be full spaces and hence, no two events a and b are independent and 0<|A|<p and 0<|B|<p. Since |C| is also going to be less than 0 and p is prime, p divides either |A| or |B|. if p was not prime (say, p=4), then you would only need the factors of p to divide either |A|,|B|, so, it becomes necessary for p to be a prime number. Hence, proved.
Learn more about sample space on
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