Intereting Posts

Does the fact that $\sum_{n=1}^\infty 1/2^n$ converges to $1$ mean that it equals $1$?
Show an exponential function has a valid density.
Commutativity of iterated limits
In which case $M_1 \times N \cong M_2 \times N \Rightarrow M_1 \cong M_2$ is true?
If $f^3=\rm id$ then it is identity function
Proving there are infinitely many pairs of square-full consecutive integers
Proof by Strong Induction: $n = 2^a b,\, b\,$ odd, every natural a product of an odd and a power of 2
How Find the $f(x)$ such $\lim_{x\to 1^{-}}\frac{\sum_{n=0}^{\infty}x^{n^2}}{f(x)}=1$
Determine: $S = \frac{1}{2}{n \choose 0} + \frac{1}{3}{n \choose 1} + \cdots + \frac{1}{n+2}{n \choose n}$
What is the difference between linear and affine function
Derivation of binomial coefficient in binomial theorem.
HCF/LCM problem
Can I switch to polar coordinates if my function has complex poles?
Inductive Proof for Vandermonde's Identity?
Range conditions on a linear operator

So what I decided to do is to start a small case with only 3 people. There are three possible combinations that I could pair up the people, using the symbols A, B, and C to represent the persons involved: A&B, A&C, B&C. The probability of each pair having the same birthday is $1\over365$ (the first guy gets any birthday, and the second guy only gets one birthday to choose from). What my intuition directed me to do next is to find the probability that any of the three events are true, or the union of the three events, which involved adding up the probabilities.

To put this generally, the equation for the probability SHOULD be:

${n \choose 2} \div 365$

- Expected maximum absolute value of $n$ iid standard Gaussians?
- Probability that one random number is larger than other random numbers
- How to calculate the expected value when betting on which pairings will be selected
- Odds of Winning the Lottery Using the Same Numbers Repeatedly Better/Worse?
- Bounds for the maximum of binomial random variables
- Sum of matrix vector products

However, clearly this is not the case since $n \choose 2$ gets over 365 when $n = 28$. And with 28 people, there is obviously still a chance that they all have different birthdays (Person 1 has Jan. 1, Person 2 has Jan. 2. … Person 28 has Jan. 28). Could anyone tell me what’s wrong with my intuition? I don’t want to know what’s the right solution, I just want to know what’s wrong with my solution .. it makes sense to my intuition, even though it’s incorrect.

- Understanding dependency graph for a set of events
- How to prove Boole’s inequality
- Using expected value to prove that there is a line intersecting at least 4 of the circles
- Odds of winning at minesweeper with perfect play
- name for a rational number between zero and one?
- Geometric or binomial distribution?
- Probability question with interarrival times
- Is this urn puzzle solvable?
- Intuition behind using complementary CDF to compute expectation for nonnegative random variables
- Proving the sum of two independent Cauchy Random Variables is Cauchy

You can only add probabilities if the events are **mutually exclusive**. But if A and B have the same birthday, it’s possible that B and C also have the same birthday. No mutual exclusivity, no addition.

You need the inclusion-exclusion principle: in your example of three people you would need to subtract the probability of all three having the same birthday to avoid double counting that event.

So the probability that there is a shared birthday is $${3 \choose 2} \frac{1}{365} – {3 \choose 3} \left(\frac{1}{365}\right)^2$$

and this should be $1$ minus the probability there are no shared birthdays

$$1 – \left(\frac{365}{365}\right) \left(\frac{364}{365}\right) \left(\frac{363}{365}\right)$$

which it is.

This can be extended to larger numbers of people, but the first method gets complicated as there may be multiple shared birthdays. The second method is easier to extend.

Actually the inclusion-exclusion principle is more involved:

Let A=event that person 1 and person 2 share a birthday.

Let B=event that person 1 and person 3 share a birthday.

Let C=event that person 2 and person 3 share a birthday.

The probability of A or B or C= the probability that at least 2 people in a group of 3 people will have the same birthday

(the same day of the year, not necessarily the same year, assuming 365 possible birth dates)=

$$P(A \cup B \cup C)= P(A)+P(B)+P(C)-P(A \cap B)-P(A \cap C)-P(B \cap C)+P(A \cap B \cap C)$$

$$P(A \cup B \cup C)= \frac{1}{365} + \frac{1}{365} + \frac{1}{365} – \frac{1}{365^2} – \frac{1}{365^2} – \frac{1}{365^2} + \frac{1}{365^2}$$

$$\frac{3}{365}-\frac{3}{365^2}+\frac{1}{365^2} = \frac{3}{365} – \frac{2}{365^2} = 0.008204166 $$=

$1−\frac{365}{365}*\frac{364}{365}*\frac{363}{365}=$ the probability that at least 2 people in a group of 3 people will have the same birthday

(the same day of the year, not necessarily the same year, assuming 365 possible birth dates)

- Edge percolation on $\mathbb{Z}^2$: probability that two neighbouring vertices are connected?
- Solution to 2nd order PDE
- Prove that two distinct number of the form $a^{2^{n}} + 1$ and $a^{2^{m}} + 1$ are relatively prime if $a$ is even and have $gcd=2$ if $a$ is odd
- Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module?
- Proving ${p-1 \choose k}\equiv (-1)^{k}\pmod{p}: p \in \mathbb{P}$
- Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$
- Logic – how to write $\exists !x$ without the $\exists !$ symbol
- On the problem $1$ of Putnam $2009$
- Why isn't second-order ZFC categorical?
- Which of the following is also an ideal?
- Equivalence relations on S given no relation?
- The product of a cofibration with an identity map is a cofibration
- Hausdorff Dimension Calculation
- What is the codimension of matrices of rank $r$ as a manifold?
- In a faithfully flat ring extension, is $\operatorname{ht}I=\operatorname{ht}IS$ right?