My question is rather easy to state, and perhaps easy to answer, but I'd need a pointer in the right direction. Suppose you have a standard 52-cards, well-shuffled deck, which contains $4$ cards for each rank (the rank of a card is the 'number' it displays, from $1$ to $13$). You draw one card at a time, for $n$ times. Cards already drawn are not put back into the deck, but just set aside. I would like to have a formula for the probability $P_n(r)$ that a card of rank exactly $r$ is drawn at the exact $n$-th extraction (i.e. after the first $n-1$ cards have been set aside).
Is anyone able to help me?
Exact formula for $P_n(r)$, the probability of drawing a card of rank $r$ from a standard shuffled deck at the $n$-th draw without replacement.
card-gamesprobability
Best Answer
The probability of a $4$ on the $17$-th card is $1/13$, since each of the $13$ ranks is equally likely to be the rank chosen at that draw (since you are talking about absolute, not conditional, probability). And the answer is the same with or without replacement.
And similarly, of course, the probability of rank $r$ on draw $n$ is $1/13$, for the same reason.