I am doing an introduction to probability book and the following problem was presented.
Consider a well-shuffled deck of n cards, labeled 1 through n. You flip over the cards one by one, saying the numbers 1 through n as you do so. You win the game if, at some point, the number you say aloud is the same as the number on the card being flipped over (for example, if the 7th card in the deck has the label 7).
Let $A_i$ be the event that the $i^{th}$ card in the deck has the number $i$ written on it.The solution starts off by saying that $P(A_i)=\frac{1}{n}$. While I can understand the reasoning that since only 1 card will match the position, the probability is therefore $\frac{1}{n}$, is it not also true that the probability distribution of $A_i$ is conditioned on $A_i$? So how can we say that $P(A_i)$ is simply $\frac{1}{n}$?
For example, we cannot use binomial distribution to solve this problem because $P(No\ match)\neq(1−\frac{1}{n})^n$. Does this not show that $P(A_i)\neq\frac{1}{n}$?