Suppose a room contains $n$ people. What is the probability that at least two people share the same birthday?
Let $A$ be the probability that at least two people have the same birthday. I know that the way to solve this question is actually to find the complement of A and solve $1 – P(A^c)$. However, I'm confused on why $A^c$ is the probability that no one shares the same birthday (everyone has different birthdays), and not the probability that at most two people share the same birthday. Isn't the opposite or complement of "at least two people share the same birthday" equal to "at most two people share the same birthday?"
Best Answer
Let $D$ denote the number of days in a year, so $D=365$ or $D=366$ (or something else) depending on how you are counting (and which planet you are living on).
The probability that no one shares the same birthday is the product of the probabilities that the second person doesn't share their birthday with the first $(D-1)/D$ times the probability the third doesn't share with the first two $(D-2)/D$ and so on down the line, until we get $$ \mathbb P(\text{no common birthdays})=\frac{D-1}{D}\cdots \frac{D-n+1}{D}. $$ So the probability that at least two people share a birthday is $1$ minus this.