If you have anything to do with the poker scene in Las Vegas, the Wednesday Poker Discussion Group (WPDG) is an invaluable resource. It is composed of avid poker players that study the game with serious intent and the group is comprised of some of the smartest people in poker, including WSOP bracelet holders and people you’ve seen playing poker on TV. A recent discussion in the group involved an MGM poker promo that the casino introduced. A member of the WPDG wanted to know the probability of this particular promo jackpot being dealt.
MGM Poker Promo: The Parameters
The parameters for the MGM poker promo, as I understood them, were the following:
- You had to make at least a full house or higher
- Both of your hold cards had to play
- The river card has to play in your hand and the river card had to be the day’s particular “cash card”. For the sake of my calculations, I used the five-of-clubs (5C) as the cash card, but you can take this to be any particular card that has to come of the river.
MGM Poker Promo: The Calculations
There’s three components to this promo:
- Probability of a 5C coming on the river
- Probability of a boat or higher being the highest hand
- Probability of your hole cards and the 5C being part of that boat or higher.
- A hand ends with a five of clubs on the river = 0.01925 or about 1 in 52 hands. This is completely reasonable; in a fair shuffle, any of the 52 cards in the deck is just as likely to come on the river.
- A hands ends with a five of clubs AND is a boat or higher = 0.0005253 or about 1 in 1900 hands
At this point we are left with a set of 7 cards where the first two are our hole cards and the last one is the 5C. We can make an inference to determine whether our hole cards and the river card are part of the hand and meet the rules for the jackpot payout.
From 7 cards on a board, you can make 2,250 5-card hand combinations (in probabilistic statistics, this is known as a 7P5 problem). But when you “lock in” the first three cards (where 3P3 = 6), as we need to do to meet the promo, and can now only choose 2 from 4, you are only left with 12 combos (a 4P2 problem). As you can probably see already, we are headed to small number of hands that actually contain the three required cards we need.
We can conclude that, for the purposes of the promo, only 72 of the 5 card combinations combinations involve all 3 cards, leaving a proportion of 0.0286. We can take that proportion and apply it to the 1900-to-1 number above and we can conclude that the promo will meet all three rules that we set out for it only once in 66,433 hands. That leaves us with a probability of this jackpot hitting on any given hand at about 0.00001505 or 0.0015% which is a fairly small probability indeed!
The biggest part of these calculations that you should walk away with is that there are very few combinations that meet all the promo rules. You will still make full houses and higher and you will still hit the cash card on the river. But there will be a lot of “near misses” where either both of your hole cards are not playing and/or the cash card not being part of the winning combination you need. You now know better, but a less-educated player might stay motivated with these near-misses and keep encouraging casinos to offer the jackpot promos.