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A group of four people is said to be “interesting", if there are at most five pairs who are friends. Assume that each pair of people are friends, independent of every other pair, with probability 1/2 . Let S be the number of pairs that are friends in this group.

What is the probability that a randomly chosen group of four people is “interesting"?

My way of thought is the following and its wrong, for me there are 6 pairs in this group of 4 persons. ( 4 c 2). Then I Calculate the at most till 5 pairs. That is I sum from 1 to 5, with the combined value and probability. But it is wrong, can you help please?

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    $\begingroup$ Instead of calculating the probability of $P(X \leq 5)$ try it with $1-P(X=6)$. $\endgroup$ Feb 15, 2019 at 13:32
  • $\begingroup$ You should sum from $0$ to $5$ or use @user2974951's short-cut $\endgroup$
    – Henry
    Feb 15, 2019 at 13:38
  • $\begingroup$ An "interesting" group is one in which there exists an unfriendly pair. $\endgroup$
    – whuber
    Mar 5, 2019 at 14:44

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Another reformulation: What is the probability that a random graph on 4 vertices is complete (when each edge is choosen independently with $p=1/2$.)

So let $X \sim \mathcal{Binom}(6,p=1/2)$ be the number of edges. Then we have $$ \DeclareMathOperator{\P}{\mathbb{P}} \P(X\le 5)=1-\P(X=6)=1-\binom{6}{6}(\frac12)^6=\frac{64-1}{64}. $$

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