Reasoning
By likely, we mean greater than 50% chance.
With one person there is a 0 percent chance that you'll have two people with the same birthday.
With two people the probability that they won't share a birthday is 364 ÷ 365. The probability that they will share a birthday is therefore 1 − (364 ÷ 365).
With three people the probability that they won't share a birthday is the same as for two people, times 363 ÷ 365. So the probability that three people will share a birthday is 1 − (364 ÷ 365) x (363 ÷ 365).
Notice that with each additional person added, the probability that they shares a birthday with one of the previous persons goes up, because there are fewer "free" days remaining.
We keep adding people until the %age is greater than 50%.
When we have 23 people the %age is 50.729%.
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Puzzle 200
Each of these words can be split in half, and two letters can be placed in the middle to create a new six-letter word.
The four added pairs of letters, in order, spell an 8-letter word. What is that word?
lend
liar
pale
coin
For example:
door + ct = do + ct + or = doctor
suit + nl = su + nl + it = sunlit