Saturday, February 22, 2014

Gambler's Ruin Problem Part 4: Odds in Video Poker

Click here to see part 1 of this four week series.
Click here to see part 2 of this four week series.
Click here to see part 3 of this four week series.


Suppose that you find yourself in a city with a casino and you have $60 in your pocket. There is a concert in town that you really want to see, but the tickets cost $100. You decide that you will place $1 bets until you either reach $100 or go broke. Which casino game should you play? How likely are you to reach your goal of $100?

This week is the final week in my series on the Gambler's Ruin Problem, which I have explained again in the paragraph above. We have analyzed roulette, craps, and blackjack, finding out various odds, expected values, and success probabilities. Here is what we have so far:

GameAvg Gain Per $1 BetP of Reaching $100
Favorable Game (51-49 odds)92.6%
Fair Game (50-50 odds)60%
Blackjack-0.5¢47.8%
Craps-1.4¢28%
Roulette-5.3¢1.3%

Let's see how video poker fairs. I chose not to do Texas Hold'em or another popular version of real poker because there is so much strategy involved, and determining a universal statistic is near to impossible. If you are also considering reading players, pot odds, and the occasional bluff, it becomes extremely complicated. However, video poker is another game that is you against the house and can be played with a more concrete strategy like blackjack. Let's go over the rules.


As you can see in the picture, video poker is pretty much a slot machine, except it gives much better odds. The game of poker played on it is five card draw. The player is given five cards to start. They then are given the opportunity to either keep them all, get rid of one, get rid of two, get rid of three, get rid of four, or get rid of all of them. The cards they got rid of are then replaced, and the hand they have left is identified and the proper money exchange is made. If you are unfamiliar with the poker hands, click here. This is the list of odds the casino gives for each hand:

Royal Flush - 250:1
Straight Flush - 50:1
Four of a Kind - 25:1
Full House - 9:1
Flush - 6:1
Straight - 4:1
Three of a Kind - 3:1
Two Pair - 2:1
Pair of Jacks or Better - 1:1


As you can see, the rarer the hand, the bigger the payoff is. This creates the difficulty in choosing which cards to hold and which to discard in any given turn.

I will start by just going through the basic strategy. Then, we will look at some of the exceptions and why they are encouraged.

1. With two pair or higher, keep the cards needed for the hand and discard the rest (for instance, draw one card in a two pair, don't draw any in a flush)
2. With one pair, keep the pair and discard the rest
3. With no high cards, discard all five
4. With one high card, keep it and discard the rest
5. With two high cards, keep them and discard the rest
6. With three or four high cards, keep two. If there are two of the same suit, keep them. Otherwise, take the two with the lowest value.

Pretty simple when it comes to poker strategy. That one won't take up an entire book. Now let's look at some exceptions. The first one is the following:

If four cards are in a straight flush, discard the fifth one (but don't break a straight or flush unless it can become a royal flush)

For instance, take the following example:

7C  8C  9C  JC  JD

In this hand, the basic strategy would say to discard the seven eight and nine, but this exception tells us to discard the jack of diamonds. Let's see why. There are 52 cards in a deck, and five have been used in this hand. So, the remaining possibilities to take the place of the jack of diamonds should total 47.

Here are the possible hands and their probabilities:

P(Straight Flush) = 1/47
P(Flush) = 8/47
P(Straight) = 3/47
P(2 Jacks) = 2/47
P(Nothing) = 33/47

EV(Discard JD) = (1/47)(50) + (8/47)(6) + (3/47)(4) + (2/47)(1) + (33/47)(-1) = 79/47 ≈ $1.68

And here are the possible hands and their probabilities for keeping the jacks:

P(4 of a Kind) = (3 ways of choosing)(2/47 third jack options)(1/46 fourth jack options) = 3/1081
P(Full House) = (3 ways of choosing placement of third jack)(2/47 third jack options)(46/46 • 3/45 pair options) + (12 value choices of three of a kind)(4/47 first cards)(3/46 second cards)(2/45 third cards) = 62/5405
P(3 Jacks) = (3 ways of choosing placement of third jack)(2/47 third jack options) = 6/47
P(2 Pair) = (3 ways of choosing)(47/47 • 3/46 pair options) = 9/46
P(2 Jacks) = 1 - P(4 of a Kind) - P(Full House) - P(3 Jacks) - P(2 Pair) = 7161/10810

EV(Keep Jacks) = (3/1081)(25) + (62/5405)(9) + (6/47)(3) + (9/46)(2) + (7161/10810)(1) = 17397/10810 ≈ $1.61

As you can see, keeping the jacks has less of a payoff than this exception. It isn't a huge difference, but still one to take note of.

Here are some other exceptions:

If you have four cards in a flush, go for the flush.
If you have three cards in a royal flush, go for the royal flush.
If you have an open-ended straight, go for the straight.
If you have three straight flush cards, go for the straight flush.
If you have J 10 suited, Q 10 suited or K 10 suited, keep the 10 in addition to the high card.

All of those exceptions could be proven with the probability techniques. Using this strategy, you will have very similar odds to blackjack with about a 0.5¢ loss per turn. Most players will lose 4¢ per turn, so this puts us way ahead of the game.

GameAvg Gain Per $1 BetP of Reaching $100
Favorable Game (51-49 odds)92.6%
Fair Game (50-50 odds)60%
Blackjack-0.5¢47.8%
Video Poker-0.5¢47.8%
Craps-1.4¢28%
Roulette-5.3¢1.3%
Average Vegas Slot Machine-6.6¢0.5%

I added an average for Vegas slot machines onto the bottom of our table just to give a taste as to how bad their odds are. Of course, each slot machine varies and I'm not sure how accurate the information I found was for them. But the odds for the games we discussed are all completely based on mathematics and are a lot of fun to play around with. If you play other games in the casino, I'm sure you can determine all of the same information for the Gambler's Ruin Problem and looking for optimal strategy.

So what's the answer to the Gambler's Ruin Problem? I don't know. We took a close look at many different games, but we can never have enough information to have a solid answer. There are dozens of different games and bets out there, many of whom can't be completely analyzed with probability and game theory. But this problem is a gateway to having fun with numbers and probability to make yourself a better player at the casino and learn a bit about mathematics. I hope some of the things we've learned in the last month become useful to you next time you have sixty bucks in a city with a casino.

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