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### Birthday Probabilities

```
Date: 12/09/97 at 23:56:40
From: Fred
Subject: Probability for people's birthdays and the days of the week
and astrological signs

What's the minimum number of people you need in order for the
probability that two of them were born on the same day of the week to
be 50 percent?  And the same for the same astrological sign. If you
could help me out; I'm a little unclear with probabilities and
statistics and such, so any help would be greatly appreciated.

Thank you.
```

```Date: 12/10/97 at 12:13:54
From: Doctor Anthony
Subject: Re: Probability for people's birthdays and the days of the
week and astrological signs

We shall calculate the probability that no two have birthdays on the
same day of the week and subtract that probability from 1

For no two to share the same day, the first person has 7 choices,
the second person 6 choices the third person 5 choices and so on.

If there are n persons

7 x 6 x 5 x 4 x ....(7-n+1)     P(7,n)
Probability = ----------------------------- = --------
7^n                     7^n

The probability that at least two have birthdays on the same day of
the week

P(7,n)
= 1 - -------   > 0.5
7^n

P(7,n)
So  ------ < 0.5
7^n

Using trial and error

n = 3   P(7,3)/7^3 = 0.6122

n = 4   P(7,4)/7^4 = 0.34985

and so 4 people is sufficient to give a better than 50% chance that
at least two have birthdays on the same day of the week.

Since there are 12 different astrological signs (instead of 7 different
days), the calculation would be

P(12,n)
--------- < 0.5
12^n

Trial and error shows that

n = 7   P(12,7)/12^7   = 0.1114

n = 5   P(12,5)/12^5   = 0.3819

n = 4   P(12,4)/12^4   = 0.5729

and so 5 people are sufficient to give a better than 50% chance that
at least two have the same astrological sign

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability
High School Puzzles

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