Follow on to discussion on Birthday Paradox
I read the Scientific American article again
Please explain why they want to find the probability of NOT matching birthdays, and then subtracting that number from 1.
Why not just say $(1/365)exp(253)$
Is it because many online exponent calculators result in 0 if you compute $0.0027exp(253)$, whereas if you compute $0.99726027exp(253)$ it results in $0.49952$
The event you are interested is that among 23 people, $$A = \{\text{At least one match}\}.$$
It's easier to consider the complementary event $$\bar A = \{\text{Not (at least one match)}\} = \{\text{No match}\}.$$ We assume that the birthdays are independent of one another and that all birthdays are equally likely in a 365-day year. Then for the first person, they can have any birthday out of 365. Then next person cannot match, thus there are only 364 options left out of 365. The third person can't match, and we've used up two days, so they only have 363 options out of 365. The rest of the people's birthdays follow similarly and thus \begin{align*} P(A) &= 1-P(\bar A) \\ &= 1-P(A_1,A_2,\dotsc,A_{23})\\ &= 1-P(A_1)P(A_2)\dotsm P(A_{23})\\ &=1-\frac{365}{365}\cdot\frac{364}{365}\dotsm \frac{343}{365}, \end{align*} where the event $A_i$ is that this person does not match the previous birthdays.
If we did it the other way, then $$P(A) = P(A_{1,2}\cup A_{1,3}\cup A_{1,4}\cup \dotsb\cup A_{23,22})\tag{$\star$}$$ where the event $A_{i,j}$ is the event that person $i$ matches $j$ where $i\neq j$. This is to be calculated using inclusion-exclusion, which is cumbersome. So, we consider the complement instead.
In the previous post, it is given that the probability of of no match among the 253 combinations is $p = 99.726027$. So, by using the analogue to $(\star)$, we would have to use inclusion-exclusion among the 253 different combinations. However, using the complement gives us, analogously,
$$P(A) = 1- P(\bar A) = 1-P(A_1)\dotsm P(A_{253}) = 1- p^{253} = 0.5004777,$$ where here $A_i$ is the event that combination $i$ was not realized.