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I was reading a Wikipedia article on the birthday paradox and stumbled upon the following statement:

...the pairings in a group of 23 people are not statistically equivalent to 253 pairs chosen independently...

Could you explain what does it mean and why does it matter in this case?

Here is the quotation in its original context:

Although the pairings in a group of 23 people are not statistically equivalent to 253 pairs chosen independently, the birthday paradox becomes less surprising if a group is thought of in terms of the number of possible pairs, rather than as the number of individuals.

Tim
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vitaut
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1 Answers1

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In a group of 23 people, all pairs must involve just those 23 people: the pairs are thus mathematically (and statistically) dependent. On the other hand, 253 pairs chosen independently and randomly from the 366*365/2 possible pairs will typically involve around 100 separate people. This (strong) dependency means we cannot use simple formulas for combining probabilities.

This vague statement is in the Wikipedia article to counter the false intuition some people have that birthday collisions in small groups must be rare. It is, as the article notes, not at all rigorous.

whuber
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  • @rm There are 366 possible birthdays. The "253 pairs chosen independently" are assumed, in contrast to the 253 pairs available from the 23 selected people, to be chosen from all possible pairs of birthdays. The point is that the number of birthday matchings available among 23 people (253) is of the same order of magnitude as the number of possible birthdays (366), making it plausible that at least one matching will involve the same two birthdays. – whuber Apr 07 '11 at 17:10
  • @whuber sorry I deleted my comment because I realized what you meant. To everyone reading it, I was basically asking why he was calculating 366*365/2. – rm999 Apr 07 '11 at 17:13
  • @rm I genuinely appreciated your comment; it's always good to get such suggestions to clarify a reply. – whuber Apr 07 '11 at 17:26
  • In the [Wikipedia analysis](http://en.wikipedia.org/wiki/Birthday_problem#Calculating_the_probability) there seem to be 365 equally probable birthdays. – Henry Apr 07 '11 at 19:39
  • @Henry In light of that discrepancy I guess I had better retract my answer, then :-). – whuber Apr 07 '11 at 19:50
  • @whuber: 23 seems to be the correct answer for 341 through to 372 equally probable birthdays, so I doubt the details matter – Henry Apr 07 '11 at 22:25
  • But what about February 29th? – Adam SA Apr 08 '11 at 16:04
  • @Adam People have done the computations for actual birth frequencies and to account for the relative scarcity of 2/29. It makes little difference. After all, the chance that one of your 23 people has a 2/29 birthday is roughly only 23/365/4 = 1.6%, too small to appreciably change the chance of a birthday collision. – whuber Apr 08 '11 at 16:07