# Probability of birthdays shared in a group

## Main Question or Discussion Point

In a group of 23 people, the probability that any 2 people share a birthday is:

1 - $$\frac{365!}{342!365^{23}}$$

Why can't I just do the following?

$$(23\mathbf{C}2)(\frac{1}{365})$$

Thanks!

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tiny-tim
Homework Helper
Hi IniquiTrance!

Do you mean 1 - 23C2/365 ?

because there are "overlaps" that you aren't subtracting …

the individual events (of one particular pair not sharing the same birthday) are not independent … eg the pairs Tom and Dick, Tom and Harry, and Dick and Harry, are not independent.

Hmm, I kind of see what you're saying...

But why is say P(Tom and Harry|Dick and Harry) different than P(Tom and Harry)?

tiny-tim
Homework Helper
Yes, the probabilities are the same, but the events are different.

Thanks for your response. I'm on the verge of uynderstanding it, can you think of any other way to explain it?

I thought the definition of independence is that $$E_{i}$$ and $$E_{j}$$ are independent events so long as $$P(E_{i}|E_{j}) = E_{i}$$

and $$P(E_{j}|E_{i})= E_{j}$$

A bit confused...

tiny-tim
Homework Helper
I thought the definition of independence is that $$E_{i}$$ and $$E_{j}$$ are independent events so long as $$P(E_{i}|E_{j}) = E_{i}$$

and $$P(E_{j}|E_{i})= E_{j}$$
I prefer to write it P(Ei and Ej) = P(Ei)P(Ej).

(because, that way, you can string more than two together)

But it's much easier just to use common-sense, and to say that the three events of Tom and Dick, Tom and Harry, and Dick and Harry, sharing (or not sharing) a birthday are obviously not independent.

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