Birthday Question from Quant Misc - Probability Fundamentals III

GRE Quant Problem Solving
1

Hi, I had a question about this problem:

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Isn’t the answer to this problem
1-(365*364*363*362)/(365^4)?

The probability that no one shares a birthday is (365*364*363*362)/(365^4), so the answer is 1 minus that.

The solution here overcounts the cases where three or four people share a birthday. The actual difference is small (it’s still approximately 1.63%) but I’m pretty sure the methodology is wrong

2

I agree with you, this is an interesting phenomenon called avalability heuristic, quote https://en.wikipedia.org/wiki/Availability_heuristic

3

@sb1182838’s approach is correct. Hint: the answer must be 1 when there are at least 366 people in the room by the pigeonhole principle.