21 people arrive separately to a professional dinner. Upon arrival, each person

Question

21 people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then either sits at the table of a friend or at an unoccupied table is none of those present is a friend. Assuming that each of pairs of people are, independently, friends with probability 0.8,find the expected number of occupied tables.

(Hint: One possible approach is to define, for example, X3 to be the random variable whose value is 1 if the third person to arrive sits at an unoccupied table and 0 otherwise.)

Answer: The expected number of occupied tables is????

Explanation / Answer

The probability that the (n+1)th person is unknown to anybody already present is 0.2n and therefore 1-0.2n if they are known to somebody. In the former case the number of tables used must increase by 1 and in the latter the number of tables used increases by 0.
Now, if X is the random variable counting the number of occupied tables then X = X1 + X2 + .. + Xn + .. + X21 where Xn is the random variables for the nth entry into the venue and is 1 when nobody knows that person and 0 if somebody knows them:

E(X) = E(X1 + X2 + .. + Xn + .. + X21)

E(X) = E(X1) + E(X2) + .. + E(Xn) + .. + E(X21)

E(Xn) can be given as: E(Xn) = 1*0.2n-1 + 0*(1-0.2n-1) = 0.2n-1 [nth arrival friend is not friend with any of first n-1]

E(X) = 0.20 + 0.21 + 0.22 + ... + 0.220

= (1-0.221)/(1-0.2)

= (1-0.221)/0.8

= 1.25 (apx)

So, we expect between 1 and 2 tables to be occupied.

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