What's the probability of at least one of them will be depressed if their friendship suddenly ends? > In a community, everyone has $p(k)$ probability to have exactly $k$ friends. Everyone with less than $K$ friends is depressed. We pick out a pair of friends. What's the probability of at least one of them will be depressed if their friendship suddenly ends? Note: They could be depressed before the "break up". One can be depressed if the number of his friends is $\leq K$, so it's probability is $$P=\sum_{i=1}^{K} p(i)$$ So the answer is $$2P=2\sum_{i=1}^{K} p(i)$$ Is this the right answer?

Persons A and B are guaranteed to have at least one friend. If one of these persons is to be depressed after the breakup, at least one of them must have a number of friends between 1 and K. That probability is:

$$P(A)=P(B)=\sum_{i=1}^{K} p(i)$$

The probability of interest to us is

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)=$$

$$2\sum_{i=1}^{K} p(i) - \left(\sum_{i=1}^{K} p(i)\right)^2$$