Not understanding division in Birthday Paradox

Question

I am reading Scientific American's explanation for birthday paradox here

I understand everything in the article up to

Every one of the 253 combinations has the same odds, 99.726027 percent, of not being a match. If you multiply 99.726027 percent by 99.726027 253 times, or calculate (364/365)253, you'll find there's a 49.952 percent chance that all 253 comparisons contain no matches.

When I compute $364/365 \times 253$, the answer is $252.306$

Why is the article saying it is $49.952%$?

Answer 1

"Multiplying 99.726027 percent by 99.726027 percent 253 times" means $$(0.99726027)^{253} \approx 0.49952.$$

Answer 2

Comment: This may be a useful intuitive argument to convince skeptics that the probability is less than half. Especially so, if the skeptic is thinking he/she is unlikely to be involved in a birthday match. However, more precisely, $P(\text{No match}) = \prod_{i=0}^{24}(1 - \frac{i}{365}) = 0.4927,$ assuming that 23 subjects are sampled at random from among 365 equally likely birthdays. The Scientific American argument does not deal appropriately with the possibility of three or more people with the same birthdays.

Another faulty line is thinking is that there can be no match with one person and that it would require 366 subjects to insure a match. Then to think that the relationship between number of subjects and a match must be roughly linear for 0 to 365. The distinctly nonlinear relationship is shown in the figure below (from R).

 n = 1:60; p  = numeric(60)
 for (i in n) {q = prod(1 - (0:(i-1))/365)
   p[i] = 1 - q}
 plot(n, p, pch=19, ylab="P(Match)")
 p[60]
 ## 0.9941227