Expanded Deck of Card Question

#1
I have been given a question by my work. It feels like a doozy to me, but hopefully it's easier than I assume. Any help with equations would be greatly appreciated.

I have been asked to find the expected payout of a promotion that centers around poker hands.

The expected payout equation are quite simple and is the summation of:

Odds of receiving a Poker Hand * The number of possible hands * Value of the Hand

For all of the various tiers of poker hands.

This is not the hard part of the equations. First, they want to have 15,000 cards with an uneven distribution. (i.e. only 5 of each of the 4 aces, a lower amount of quantity for specific other cards, but a relatively even distribution beyond that.) This is still not all that hard as it just changes the number of possible hands for 4 of a kind, flush, royal flush, straights, etc.

The twist is that all of the cards will be given out. (Some people will get more cards, others less, but all cards can be passed out. It's assumed some people will trade/give cards, but it won't be rampant as trading between strangers won't happen.)

The numbers that I need to turn into my boss have to account for the chances that if a Royal Flush is redeemed, then the odds of a 4-of-a-Kind for Aces will go down. (And other such instances.)

Is there a way to account for this in my equations? Or is this accounted for by the standard equations?

I'm sorry that I am not using all of the standard terminology. I'm about 10 years out of my last math class! :)

Thank you for your help in advance!
 

Mean Joe

TS Contributor
#2
Seems like you just need to calculate the probability of getting 4 Aces, given that there is a Royal Flush.

So I'm calculating 14,991 out of C(14995, 5),
where C(14995,5) = 14995!/(5!14990!)