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In a room of 23 people there is a 50.7% chance that two share a birthday, because the room contains 253 possible pairs rather than 23 chances to match
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With 23 people, there are 253 possible pairs.
Person one can match person two, three, four and everyone after them.
Person two can match person three, four and everyone after them.
We cannot simply add 1/365 across all 253 pairs because the events overlap.
Once the no-match probability falls below one-half, the match probability rises above it.