The birthday paradox has returned. And that too with vengeance. Please keep up with me for this time my question is somewhat "lengthier" this time

In a recent book by keith ball, i read some beautifully wonderful demonstrations of many mathematical truths which i believe were never earlier published at a popular level. The book is called

"Strange Curves, Counting Rabbits, & Other Mathematical Explorations".

Now back to the problem.

The author discusses at length the strange feeling you feel after your first encounter. He proceeds by telling us to find the expectation of getting a match with k ppl when we assume that there are n days in an year.

(A match is simply a pair of persons sharing the same b'day)....

Let us assume that there are 100 persons for the time being i.e k=100
A paraphrase of what Ball says is given below

"There are several ways to calculate expected number of matches; the avg num of matches we would get if we kept taking groups of 100 & running the experiment.

One way is to calculate the chance of getting exactly one match P1, the chance of getting 2 matches P2, & so on and form tne sum

P1 + 2.P2 + 3.P3 + ---- + m.Pm <======== NOTE THIS

But obviously this process is going to earn lots of hate-mails.

An easier way is to notice that the chance that a particular pair of people share their b'day is 1/n..

Also num of possible pairs is K.(K-1)/2...So, expected number of matches is simply X = K.(K-1)/2n..

So far so good...Now the trouble starts.

For Ball writes

"Suppose n events occur with probabilities P1, P2, P3...Pn respectively and that these probabilities are rather small. Show that if the events are independent, then the chance that none of them happen is roughly e^(-k).. where k is expected number of happenings given by

K = (SUM i:= 1 to n) Pi" <========= NOTE THIS

...This is what i find troubling, for earlier Ball write that the expectation, X = P1 + 2.P2 + 3.P3 + ------ + m.Pm

And now he assetes that X = P1 + P2 + P3 + ---- + Pn..

Well, i understand that in the first equation we are interested in number of matches, m and in the next we talk of number of persons, n..

Also, i notice the Pi's in 2 equation stand for different things, but what i question is - how come these two seemingly different expressions point at the same value for expectation... (that the first one does is clear too...but i can't say the same for the second one..and certainly not for them both taken together)..

Thank You and Have a nice Day
-Akash Kumar

Einstein - You know, earlier my preference mathematics. Later on i changed it to physics

Pincare: And why is that?

Einstein: I could not tell important facts from non-important ones.

Poincare: Starne that you say so. But now that you mention it, earlier i cherished physics..Now i cherish mathematics.

Einstein: And why is that