So the birthday paradox is where if you’re in a room with 23 other people there’s a 50% chance of at least two people having the same birthday. Alternatively, In a room of 75 there’s a 99.9% chance of at least two people matching.
Why is this?
In: 134
Anybody I meet with the same birthday must be willing to fight to the death. Otherwise they must be comfortable with lying to me about when their birthday is. Love you homie.
The logic behind the probability of two people sharing a birthday being 1/365 is thus:
Pick a date at random(say March 1). What are the odds that person A birthday is March 1? 1/365. What are the odds of person B having a birthdate of March 1? 1/365. Odds of A and B having birthdate March 1?
P(A and B) = (1/365)* (1/365)
Now the question isnt about a particular date, but rather if they share a birthday regardless of the date. So we need to calculate the probability of not just March 1 but every possible date in the calendar(not assuming leap year), which is 365. So the probability becomes 365 times P(A and B) which is:
= 365* (1/365)* (1/365)
= 1/365.
There you go
It’s only a *veridical* paradox, which just means that it defies the natural assumptions our brains make.
Briefly, lets look at a coin flip. What are the odds we *don’t* get heads? There is:
0.5 chance to not get heads once.
0.5*0.5=0.25 chance to not get heads twice.
0.5^n to not get heads n times.
We get a basic formula from this for how probability changes for the number of samples. If p is the probability and n is the number, then the formula is p^n
For the birthday paradox, we use the same tactic, but instead ask the odds of two people *not* sharing a birthday.
Because the people are random, each pair has the exact same chance to *not* share a birthday, 364/365.
We can form 253 *unique* pairs of people from a group of 23. If we name them A-W, A has 22 possible pairs, B has 21 since we don’t want to count A again, and this goes on for each person.
So our formula is:
(364/365)^253 = 0.4995 chance they *don’t* share a birthday. This means:
0.5005 chance they *do* share a birthday.
**The Math:**
(364/365)^253 = 0.4995
**The Explanation:**
253 unique pairs come from 23 people.
364/365 chance they *dont* share a birthday.
Like a coin flip, each time we check we are actually multiplying the odds exponentially. With a coin flip the probability is 0.5, but here it is 364/365.
So the result, 0.4995, is the odds they don’t share a birthday. That means the odds they *do* share a birthday is 0.5005, or 50.05%
It is a veridical paradox, which just means it defies our brains’ natural assumptions. It isn’t a paradox in the traditional sense.
Can we have an explanation like I’m 2?
My new girlfriend and I have the same birthday, and one of her ex also has the same birthday. Thats a small percentage in odds! (Or she slept with 77 guys in a room… hmm)
Latest Answers