I've been wondering how to compute this one for a while, and I've done some initial work but it seems to be going nowhere.

Find the probability of holding a Royal Flush in a n hand draw, where n must be atleast 5 but no greater than 52. Also, find the value of n that gives a 100% chance to draw a royal flush.

I have seen some VERY obscure stuff about this question online, but I'd like to get down to business on it!

Edit 3: You wrote "the probability of holding a Royal Flush in a n hand draw," but I suppose you meant an n-card hand? Because why else would n range from 5 to 52. Or maybe it's just because I'm unfamiliar with poker terminology. (I moved this edit to the top because my whole post is based on it.)

Well there are only 4 royal flushes, so that should help keep the numbers manageable.

I'll tackle the last question first: The probabilty of getting a royal flush is 1 if and only if \(\displaystyle n > 48\). This is because at least one of the five cards from each royal flush must remain in the deck, so for example, if you get a 48 card hand but all 4 aces are left in the deck, then you won't have a royal flush, and this is the most cards you can have in your hand for that to happen.

I'll have to think about the other stuff. Someone will probably post an answer before I get the chance to sort it all out.

Edit: Well the probability of getting a royal flush in a 5-card hand is \(\displaystyle \displaystyle \frac{4}{\binom{52}{5}}\).

Edit 2: For n = 6, we have 4 ways to choose a flush, and then 52-5=47 ways to choose the remaining card for each flush. So \(\displaystyle P_6=\displaystyle\frac{4\cdot47}{\binom{52}{6}}\)

And likewise \(\displaystyle P_7=\displaystyle\frac{4\cdot\binom{47}{2}}{\binom{52}{7}}\)

I think starting at n=10 we'll have to use inclusion-exclusion.