Mind Bender # 10 : Solution

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Last month only two people managed to come up with the correct solution, as well as the reason why. Therefore, I held back their postings until now.

The answer is that you should certainly take the odds of 30:1 on offer — because your chances of winning the wager are considerably better than that.

Anyway, for those of you still puzzling over the chances of two people having their birthday on the same day in the year, in a room of 23 unrelated people … here’s how you work it out.

h3. You just need to calculate things in reverse!

The probability of you NOT having your birthday on the same day as the 1st person in the room would be … 364/365 (ie: a 99.73% chance of it NOT occurring).

Therefore, NOT having your birthday on the same day as the 2nd person would be … 363/365 (ie: 99.18%)

And so on, down to … 343/365 (ie: a 93.97%) chance of you NOT having it on the same day as the 22nd person.

And the combined probability of nobody having their birthday on the same day as each other … is found by multiplying all these individual possibilities together.

As you’ll see set out in the Table, the probability of NONE of the 23 people having their birthday on the same day is 49.27%.

Therefore (by subtraction), you would have a 50.73% chance of at least two people having their birthday on the day.

In other words … you actually have an “Even Money” chance of winning the wager.

So, you should grab those outrageous odds of 30:1 with both hands. And just remember this little party trick for later.

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