Question

what is the average number of pairs of consecutive integers in a randomly selectedsubset of 5 distinct integers chosen from the set{1,2,3,...,30}? (for example the set{1,17,18,19,30}has 2 pairs of consecutive integers.

Answer 1

the average number of pairs of consecutive integers in a randomly selected subset of 5 distinct integers chosen from the set{1,2,3,...,30} is 2/3

There are 29 possible pairs of consecutive integers namely p1 = { 1,2} ....... ,p29= {29,30}

define a random variable Xi with Xi =1 if pi is a part of the 5 elements subset and 0 otherwise then the number of pairs of consecutive integers in a 5 selection is given by the sum X1+ .........+X29

E[ Xi]= 28C3/ 30C5

        = 2/3

Hence ,  the average number of pairs of consecutive integers in a randomly selected subset of 5 distinct integers chosen from the set{1,2,3,...,30} is 2/3

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