###### Asked in Probability

Probability

# What is the smallest number of people in a room where the probability of two of them having the same birthday is at least 50 percent?

## Answer

###### Wiki User

###### September 17, 2007 4:46AM

23. The probability that at least two people in a room share a
birthday can be expressed more simply, mathematically, as 1
*minus* the probability that *nobody* in the room shares
a birthday.

Imagine a fairly simple example of a room with only three people. The probability that any two share a birthday is :

1 - [ 365/365 x 364/365 x 363/365]

*i.e. 1-P(none of them share a birthday)*

=1 - [ (365x364x363) / 3653 ]

=0.8%

Similarly,

P(any two share a birthday in a room of 4 people)

= 1 - [ 365x364x363x362 / 3654 ] = 1.6%

If you keep following that logic eventually you get

P(any two share a birthday in a room of **23** people)

=1 - [(365x364x...x344x343) / 36523 ] = 51%

## Related Questions

###### Asked in Probability

### If 15 strangers are all in a room what is the probability of them all having the same birthday?

To determine the probability of 15 random people all having the
same birthday, consider each person one at a time. (This is for
the non leap-year case.)
The probability of any person having any birthday is 365 in 365,
or 1.
The probability of any other person having that same birthday is
1 in 365, or 0.00274.
The probability, then, of 15 random people having the same
birthday is the product of these probabilities, or 0.0027414 times
1, or 1.34x10-36.
Note: This answer assumes also that the distribution of
birthdays for a large group of people in uniformly random over the
365 days of the year. That is probably not actually true. There are
several non-random points of conception, some of which are spring,
Valentine's day, and Christmas, depending of culture and religion.
That makes the point of birth, nine months later, also be
non-uniform, so that can skew the results.

###### Asked in Chemistry, Statistics, Periodic Table, Probability

### What is the probabililty of at least 2 people same birthday from a group of 12 people?

The probability of at least 2 people in a group of n
people sharing a common birthday can be expressed more easily
(mathematically) as 1 minus the probability that
nobody in the group shares a birthday. Consider two people.
The probability that they don't have a common birthday is 365/365 x
364/365. So the probability that they do share a birthday is
1-(365/365 x 364/365) = 1-365x364/3652 Now consider 3 people. The
probability that at least 2 share a common birthday is
1-365x364x363/3653 And so on so that the probability that at least
2 people in a group of n people having the same birthday =
1-(365x363x363x...x365-n+1)/365n = 1-365!/[ (365-n)! x 365n
]
In the case of 12 people this equates to 0.16702 (or 16.7%).

###### Asked in Statistics, Birthdays, Probability

### What is the probabililty of at least 2 people same birthday from a group of 13 people?

19.4%
CALCULATION:
The probability of at least 2 people having the same birthday in
a group of 13
people is equal to one minus the probability of non of the 13
people having the
same birthday.
Now, lets estimate the probability of non of the 13 people
having the same birthday.
(We will not consider 'leap year' for simplicity, plus it's
effect on result is minimum)
1. We select the 1st person. Good!.
2. We select the 2nd person. The probability that he doesn't
share the same
birthday with the 1st person is: 364/365.
3. We select the 3rd person. The probability that he doesn't
share the same
birthday with 1st and 2nd persons given that the 1st and 2nd
don't share the same
birthday is: 363/365.
4. And so forth until we select the 13th person. The probability
that he doesn't
share birthday with the previous 12 persons given that they also
don't share
birthdays among them is: 353/365.
5. Then the probability that non of the 13 people share
birthdays is:
P(non of 13 share bd) =
(364/365)(363/365)(362/365)∙∙∙(354/365)(353/365)
P(non of 13 share bd) ≈ 0.805589724...
Finally, the probability that at least 2 people share a birthday
in a group of 13
people is ≈ 1 - 0.80558... ≈ 0.194 ≈ 19.4%
The above expression can be generalized to give the probability
of at least x =2
people sharing a birthday in a group of n people as:
P(x≥2,n) = 1 - (1/365)n [365!/(365-n)!]

###### Asked in Birthdays, Probability

### What is the probability that out of 40 students none of them will have a birthday in March?

3 % No, not correct.
------------------------------------------------------------------------------------------------
The probability that a single person would have a birthday in
March is 1 out of 12 (because there are 12 months in the year).
Hence the probability that one of 40 students would have a birthday
in March is 40 x 1/12 = 10/3 = 33.33%; More accurately March has 31
days out of 365 days min the year so the probability of one person
having a birthday in March is 31/365, and for 40 students it would
be 4 x 31/365 = 124/365 = 0.3397(to 4 decimal places) = 34% to
nearest 1%

###### Asked in Demographics, Birthdays, Probability

### How many people have the same name and birthday as you?

The first thing to note is that names and birthdays are
independent of each other. Someone born on 5th June isn't more
likely to be called Chris, and someone called Katie isn't more
likely to be born on 20th October, or whatever.
Thus the probability is equal to the probability someone has
your name, multiplied by the probability someone has your
birthday.
The latter is just 1/365.25, as you are equally likely to be
born on each day.
The former could be anything. If you have a really common name,
that 1 in 1,000 people have, then the probability of someone having
the same birthday and name as you will be 1/365,250
Multiply this probability by the world population (which we'll
round to 7 billion) and you get:
7,000,000,000/365,250
= 19,165
so 19,165 people would have the same name and birthday as
you!

###### Asked in Probability

### What is the probability that at least 2 students in a class of 36 have the same birthday?

About 83.2%
The probability that non of the 36 students have the same
birthday (not considering
February 28 of the leap year) is given by the following
relation:
P(non out of n have same bd) = Π1n-1 [(365-i)/365]
P(non out of 36 have same bd) = (364/365)(363/365)(362/365) ...
(331/365)(330/365) =
= 0.167817892.. ≈ 16.8%
So the probability of at least 2 having the same birthday is
about 1 - .168 = 0.832 =
83.2%

###### Asked in Labor and Birth, Birthdays, Probability

### What are the odds of having kids with the same birthday but a different year?

In probability theory, the birthday problem, or
birthday paradox[1] pertains to the probability that in a
set of randomly chosen people some pair of them will have the same
birthday. In a group of 10 randomly chosen people, there is an
11.7% chance. In a group of at least 23 randomly chosen people,
there is more than 50% probability that some pair of them will both
have been born on the same day. For 57 or more people, the
probability is more than 99%, and it reaches 100% when the number
of people reaches 367 (there are a maximum of 366 possible
birthdays). The mathematics behind this problem leads to a
well-known cryptographic attack called the birthday attack. See
Wikipedia for more:
http://en.wikipedia.org/wiki/Birthday_paradox