Calculating the Probability of a Shared Birthday among n People

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In summary, the probability that at least two people in a room of n people have a common birthday can be calculated by subtracting the probability that all n people have different birthdays from 1. This can be done by multiplying the probabilities that each consecutive person has a different birthday and subtracting that from 1. Therefore, the probability can be represented as 1 - (((365-(n-1))*(365-(n-2))...(364))/(365^n)).
  • #1
buddingscientist
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Suppose a room contains n people. Assuming that days of the year are equally likely to be birthdays for each person, calculate the probability that at least two of the people have a common birthday.

well I have the answer but I'm just curious as to the thought processes you go through to answer it.


what I've done (incorrectly) is:

if n = 2, P = 1/365^2
if n = 3, P = 1/365^2 (person 1, 2) + 1/365^2 (person 1, 3) + 1/365^2 (person 2, 3)

and generalized it to:
( 3 + (n sum i=3) (i)! / (i-1)! ) / 365^2

there's no question that it's wrong but that's the process I take
 
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  • #2
work out the probability they all have different birthdays and subtract that from 1. Obviously if there are 366 people in the room the probabilit must be 1, mustn't it? (excludeing leap year things). Does n=366 give P=1 in your answer?
 
  • #3
probably not (npi), I know my answer is wrong, you recommend using conjugates (1 - P(all different)), I'll work on it some more, thanks for your response
 
  • #4
Ignoring leap years:

P(bd2 = bd1) = 1/365
P(bd2 != bd1) = 364/365
P(bd3 = bd1) + P(bd3 = bd2) = 2/365, given that bd1 != bd2
P(bd3 != bd1) AND P(bd3 != bd2) = 363/365

Therefore P(bd1 != bd2 != bd3) = 1 - (363/365) * (364/365)

So for any n,
P(two birthdays are the same) =
1 - (((365-(n-1))*(365-(n-2))...(364))/(365^n))
 

What is probability?

Probability is a measure of the likelihood that a certain event will occur. It is usually expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

How is probability calculated?

Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This can be represented as a fraction, decimal, or percentage.

What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. Experimental probability is based on actual results from an experiment or observation, and may differ from theoretical probability due to random chance.

What is the difference between independent and dependent events?

Independent events are events where the outcome of one event does not affect the outcome of the other. Dependent events are events where the outcome of one event is affected by the outcome of the other.

How can probability be used in real life?

Probability is used in many real-life situations, such as predicting the weather, calculating insurance rates, and making financial decisions. It can also be used to analyze data and make predictions in fields such as genetics, economics, and sports.

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