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Asked By :  Romesh
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Assuming that birthdays are uniformly distributed on 365 days

Assuming that birthdays are uniformly distributed on 365 days of the year, then what is the probability that 25 randomly selected people will all have different birthdays?




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To find the probability that 25 randomly selected people all have different birthdays, we assume:

  • Each birthday is equally likely (uniform distribution across 365 days),

  • No leap years (i.e., exactly 365 days),

  • No shared birthdays among the 25 people.


Step-by-Step Calculation:

This is a classic Birthday Problem scenario.

The total number of possible birthday assignments for 25 people (with possible repeats) is:

36525365^{25}

The number of favorable outcomes (no shared birthdays) is:

365×364×363××(36524)=365!(36525)!365 \times 364 \times 363 \times \cdots \times (365 - 24) = \frac{365!}{(365 - 25)!}

So the probability that all 25 people have different birthdays is:

P=365×364××34136525=365!(36525)!36525P = \frac{365 \times 364 \times \cdots \times 341}{365^{25}} = \frac{365!}{(365 - 25)! \cdot 365^{25}}


Approximate Value:

Using a calculator or software to compute this:

P0.4313P \approx 0.4313


Answer: ≈ 0.4313 (or 43.13%)

So, there's about a 43.13% chance that all 25 people have different birthdays. Want to see how that changes with 23 or 30 people?



Answered By

Romesh

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