Let the sample space be the set of positive integers and suppose that P(n)=1/2^n

Question

Let the sample space be the set of positive integers and suppose that P(n)=1/2^n for n=1,2..... Find the probability of the set (3,6,9....) that is of the set of positive integers that are multiples of 3
Let the sample space be the set of positive integers and suppose that P(n)=1/2^n for n=1,2..... Find the probability of the set (3,6,9....) that is of the set of positive integers that are multiples of 3
Let the sample space be the set of positive integers and suppose that P(n)=1/2^n for n=1,2..... Find the probability of the set (3,6,9....) that is of the set of positive integers that are multiples of 3

Explanation / Answer

probability of the set (3,6,9....) =P(3)+P(6)+P(9)+P(12)+P(15)+....

=(1/2)3+(1/2)6+(1/2)9+(1/2)12+(1/2)15+.......

( this is geometric distribution for which first term a =(1/2)3 ; and common ratio =(1/2)3 whose sum

=a/(1-r))

hence probability =(1/2)3/(1-(1/2)3) =(1/8)/(1-1/8) =(1/8)/(7/8)=1/7

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