Chapter 3: Variability - Heaven and Hell

“I was losing all night, and then I got four aces twice in five minutes, which put me ahead. What are the odds of that happening?”

“This machine had to be rigged, I played four hours without getting a four-of-a-kind.”

“I held an ace of diamonds, I hit the draw button, and the machine lights up with a royal - it was fantastic”.

Anybody who has played video poker has stories of great wins and great losses. In a sense, that’s the heart of the thrill -- a roller coaster is fun because it goes up so high and comes down so fast. Who wants to ride a coaster that rolls down a slight decline? In addition, it’s great fun to share the great successes and the tragic failures with others.

When you play video poker, the highs can be dizzying, and the lows can be quite painful. It’s very important to understand what you’re getting yourself into before you end up with more pain than you can deal with.

In order to understand the ups and downs of video poker, we need to first understand some basics. First, each hand of video poker is random, which means that each of the cards has an equal chance of being dealt. It’s like flipping a coin, in that heads and tails both have an equal chance of coming up. In video poker, a computer chip does the choosing of which cards get drawn. A crucial part of this randomness of the deal is that each hand is independent of any previous hands dealt.

The human mind is not naturally suited to understanding the concept of the independent random events. In fact, the ability to predict future events from past events is crucial to the survival of almost all living things. But in video poker, this wonderful ability that we have can be a serious liability.

To illustrate, let’s look at an example.

Imagine that you’ve just hit a royal flush. Do you feel like you should quit for the night because there’s no way that you are going to hit another royal flush? You think to yourself -- “What is the probability of hitting two royals in one night?”. In truth, it does not matter at all that the previous hand was a royal, the next hand still has the same chance of being a royal.

Let’s look at another more dangerous example. You’ve been playing for seven hours and you have not hit one four-of-a-kind (for those who do not play, this represents a very, very bad streak of luck). Do you think that you are more likely to hit a four-of-a-kind now? Despite the strong feeling you have in your gut that you are “due”, your chances of hitting a four-of-a-kind are just what they have always been and just what they will always be. The probabilities do not care about your gut feelings, but you need to be aware of them because they can lead to errors in judgment that can be costly.

So how do we know how high the highs should be and how low the lows should be? Fortunately, math nerds like myself have worked out the probabilities for video poker. The best way in which these probabilities are understood is with the assistance of a picture. Specifically, a distribution shows us how likely each kind of outcome will be. For example, look at the distribution below (Figure 9) for playing 5000 hands of Deuces Wild poker with expert strategy. A moderately fast player can easily play 5000 hands in one day by playing 625 hands per hour for 8 hours.

Figure 9

The shape of the curve is like a bell. This shape approximates a “normal” curve, which is a standard distribution for random events.

The position on the x-axis (from left to right) is the gain or loss.

The position on the y-axis (from bottom to top) is the probability of that particular outcome occurring. The absolute numbers on the y-axis are not important, the overall shape of the curve is the important thing.

For example, in the distribution above, the most likely outcome is to end up with a loss of about $150 dollars. This corresponds to the peak (or hump) of the graph. How can the most likely outcome be a $150 loss when you are playing a game in which you are expected to win? The answer lies in the fact that the curve is skewed (or unbalanced) with the right side of the curve extending further out than the left side of the curve.

This skewing of the curve happens because there is a greater chance of winning a lot than losing a lot. If you look on the curve, you can see that the chances of losing $1000 is virtually nil (because the height of the curve at -1000 is so low). In contrast, the chances of winning $1000 are much higher (because the height of the curve at +1000 is a little less than 0.01).

This skewing is much more evident for smaller numbers of hands. While 5000 hands sounds like a lot of video poker, the distribution only becomes more balanced (i.e. not skewed) when you have played a few hundred thousand hands.

While the most likely outcome is to lose $150, this is not the same as saying that you will lose $150 on the average. On the contrary, for 5000 hands of Deuces Wild, you will win about $45 (5000 X $1.25 X .7%) on the average. Let’s look at an example where we examine 5 sessions of 5000 hands. What usually happens is that you might lose three times and only win twice. However, the size of your wins will be much larger than the size of your losses. Overall , these larger wins will more than compensate for your smaller losses and you will end up ahead.

We can use the distribution to answer any question about the probability of an outcome. Let’s look at how we would answer the following questions.

“What is the probability of being ahead?”

“What is the probability of being more than $1000 ahead?”

“What is the probability of being more than $1000 behind?”

To answer these questions, we look at the area under the curve to determine the answer. The area under the entire curve represents the total probability of 100%.

To answer a particular question, we look at the area under the curve that corresponds to the question and compare it to the total 100%.

Let’s look at the question of being ahead for this 5000 hand distribution.

The area under the curve that corresponds to being ahead is the area to the right of the vertical line that starts at 0 (this line roughly splits the distribution in half). Because the area to the right of this line is about 44%, there is a 44% chance of being ahead in this game after 5000 hands.

Likewise, the area under the curve that corresponds to being ahead more than $1000 is the area to the right of the line at a gain of $1,000. Because the area to the right of this line is about 7%, there is about a 7% chance of being ahead more than $1,000 after 5000 hands.

With this distribution, you can answer any question about probability by looking at the area under the curve. The ends of the curve drop off to near 0, so you don’t have to worry about points to the far left (big losses) or far right (big wins).

So what does this distribution tell us? It tells us that video poker is very risky in the short term, even if you are playing a positive reward game. In one day of playing, there is a 30% chance that you’ll lose more than $200 and about a 12% chance that you’ll lose more than $400. Remember, that these statistics are for Deuces Wild, which is a high-paying video poker game which actually offers a positive reward.

In the short run, luck will mean a lot more than your choice of game or your skill level. To illustrate this point, look at the two distributions below in Figure 10. One distribution represents the full-pay version of Deuces Wild (return = 0.7%) and the other represents a lesser-pay version of Deuces Wild (return = -1.3%). As you can see, these distributions don’t look all that different, even though one game is a smart play and the other is not.

Figure 10

OK, it’s clear what can happen in a day of play. But what if we look at what happens over longer periods of time? Look at Figure 11 which shows the distributions for 20,000 hands (a long weekend of video poker).

Figure 11

With more playing, two things happen which are important to point out.

The good news is that the distribution for the full-pay shifts to the right (e.g. towards gains) as the average expected gain increases. This means that your overall expected gain increases with more play. It also means that your chances of being ahead overall increase as well. This occurs because the skewing that happens with smaller numbers of hands disappears and the distribution becomes more balanced.

The other news is that the distribution widens to include greater losses and gain. This means that, along with greater chances of being ahead, there is an increase in the chance that you will behind a lot. For example, as can be read from the distributions, the probability of being more than $1000 behind (i.e. the percent of the area under the curve that is to the left of -$1000) after 5000 hands is less than 1%. However, the probability of being more than $1000 behind after 20000 hands rises to about 7%.

In the long run, the game you choose to play will make a great difference. The longer you play, the more difference it is likely to make. What’s the long run? Here, the long run represents playing over a period of months or even years (depending on how much you play in each video poker outing). Figure 12 shows what happens after 200,000 hands when you play full-pay Deuces Wild and when you play the short-pay version of Deuces Wild. As you can see, the distributions are pretty separate. The probability of being ahead is about 72% in the full pay game, but only about 28% in the short-pay version. You may be wondering why this distribution isn’t much flatter than the distribution when you play 20,000 hands (Figure 11). In reality, this distribution is flatter, but if you look closely, you will notice that the scale of the graph has changed. This graph extends from -10,000 to 10,000, while the graph for 20,000 hands (Figure 11) extends from -3,000 to 3,000.

Figure 12

Like good machine selection, your playing skill will make little difference in the short run, but a lot of difference in the long run. Figure 13 shows how three levels of skill (Poor=-2.8%, Good= -0.7%, Expert = +0.7%) can expect to perform in a couple hours of play (1000 hands) for Deuces Wild.

Figure 13

As you can see, the skill level does not make such a big difference for a few hours of play. It is quite likely that the bozo playing next to you might win way more than you do. If this occurs, take heart, because in the long run, things will look very differently.

Figure 14

Figure 14 shows these same levels of skill with 20,000 hands of play. As you can see, the distributions are starting to separate quite a bit. However, if you have some bad luck, you can still end up behind, and with some good luck, the bozo can still end up ahead.

Figure 15

After 200,000 hands, the distributions are quite separated. However, it is still possible (although not probable), that the bozo could be ahead of you.

Figure 16 shows the likelihood of being ahead in the three most popular games. Importantly, these graphs do not include other comps (with other comps, all 3 games will most likely put you ahead in the long run).

Figure 16

As figure 16 shows, your chances of being ahead in Deuces Wild rise slowly over time, while your chances of being ahead in Jacks or Better slowly fall over time

By now, you’ve got the message that playing video poker is never a sure thing. To hammer this home, how many hands do you think you would have to play to be 99% sure that you would be ahead? The answer is at least 2 million hands. That is about 2,500 hours of high speed play which is about a year of full-time play. So don’t quit your day job.

How much money do you need to play video poker? The answer depends on two things. First, you need to consider how long you want to play. Second, you need to consider how sure you want to be that you don’t run out of money. In order to be absolutely sure that you did not run out, you would have to bring enough money to handle the case that you lost every hand. The probability of losing every hand is so low that you can consider it to be impossible. Instead of being absolutely sure, it’s good to imagine a probability of running out that is reasonably close to 0. Figure 17 shows how much bankroll (money you bring to play with) you would need to be 90% sure that you would not run out of money.

Figure 17

As you can see, if you want to play 1000 hands (the points on the farthest left of the graph), you still need about $200 to be 90% sure of not running out. For 5,000 hands, you need about $500 for all 3 games. At 20,000 hands, you need about $1000 for all games. For 80,000 hands, you will need quite a bit more for Jacks or Better (because you are expecting a negative financial reward) and for Double Bonus (because it is pretty high in volatility). With Deuces Wild, you will need less of a bankroll in the long run because you are more likely to win.

If you want to be really sure (95%) of not running out of money, you will need to have that much additional money. Figure 18 shows the bankroll needed to insure a 95% probability of not running out of money.

Figure 18

Even if you bring enough to support a 95% certainty, that still means that 1 out of 20 trips will end in disaster. That’s OK. Just bring your limit and don’t exceed it.

So what is a typical pattern of gains and losses for a number of trips? The answer is that there really isn’t a typical set, but certain patterns are likely to occur. As stated before, if you play video poker for a weekend, then you are actually more likely to end up behind than ahead. However, when you do end up ahead, your gain from that weekend is likely to be significantly larger than your average loss.

Figure 19 is taken from twelve actual trips that a friend of mine took to Las Vegas over the course of one year. My friend plays Deuces Wild most often but will sometime play Double Bonus. In the figure, there are 3 lines that represent a running total for the year. The ‘Cash Gain/Loss’ line shows the total net loss or gain from gambling. This included any money from slot club cash back programs. The ‘Net w/travel exp.’ line shows the total position after subtracting the travel costs of going to Las Vegas. The ‘Net+Food’ line adds the cost of food that my friend did not have to pay because he got free food from the casino. It is important to note that the cost of food was estimated very conservatively. According to the graph, my friend earned about $360 worth of food in twelve trips. In actuality, it would have cost him at least twice this amount if he had paid his food bills at the casino restaurants. The food credit on the graph was reduced because when your food is comped, you eat much more expensive food than you normally do. If the actual price of the food was included in the graph (instead of about ½ the price) then this might present a slightly misleading picture. If you are a person who eats expensive food anyway, then you really will be saving more through comps. The ‘Net+Food+Room’ line adds the cost of free rooms. For these trips, my friend received about a thousand dollars in the form of free rooms. As with the food comps, there is uncertainty in how to value a free room at a casino. In general, the graph reflects a pretty conservative estimate (about $40 per night). For 12 trips, my friend won about $2200. This is close to what the expected win should be considering that he played about 240,000 hands. The expected win is $2100 because 240,000 hands X 0.7% X $1.25= $2100.

Figure 19

Out of 12 trips, 6 were winning trips and 6 were losing trips. While there are an equal number of losing and winning trips, they may occur in streaks. After losing on trips 5, 6, 7, 8, and 9, my friend won on trips 10,11, and 12. While this sample data may be somewhat more streaky than you would be likely to expect, you have to be ready for the possibility of several consecutive losing trips.

One of the reasons why more trips will be losing trips is because when you do not hit a royal, you are playing with a negative reward. For example, while Deuces Wild provides a 0.7% return, 1.7% is from the royal. This means that you will be playing with an expectation of a -1% reward (i.e. a 1% loss) when you are not getting a royal.

Importantly, the size of the gains (from winning trips) exceed the size of the losses (from the losing trips). While trips 4 and 12 both produced about $1300 wins, trip 7 produced a $900 loss which was the largest losing trip. Losing $900 in a weekend of play is a very unpleasant experience, but it is not all that uncommon if you play a lot in a long weekend. On the other hand, winning $1300 in a weekend is pretty exhilarating. In fact, in one of the $1300 trips, my friend actually hit two royal flushes. This is not a very likely occurrence, but it is not as uncommon as you might think.

How likely is it to hit a royal flush? Of course, the answer depends on how many hands you play. Figure 20 shows the distribution of royal flushes playing 5,000 hands of Deuces Wild (where the royal is expected to about every 50,000 hands).

Figure 20

As you might guess, the chances of hitting a royal in 5,000 hands are a little less than 1 in 10 (probability = .09). Many people assume that if the chances of hitting a royal are 1 in 50,000, then the chance of hitting a royal in 5,000 hands would be 5,000 in 50,000 (or 1 in 10). This is not true because there is some possibility of hitting more than 1 royal. In fact, the probability of hitting two royals in 5,000 hands is about .0045 (or about 1 in 220). If you get really lucky, you might hit three royals, but this only happens 1 in 5,000 times. Because the probabilities have to add up to 1, the probability of hitting one royal has to be less than the number of hands divided by the probability. This fact becomes more obvious when you consider what happens after 50,000 hands.

If you play 50,000 hands, the probability of hitting a royal is probably lower than you would guess. Again, you might think at first that if the chance of hitting a royal is 1 in 50,000, then this would guarantee a royal after 50,000 hands. Of course, if you seriously consider this idea, you realize how this cannot be true. In fact, as Figure 21 shows, the chance of getting no royals after 50,000 hands is exactly equal to the chance of getting one royal.

Figure 21

Again, this is true because there is about an 18% (probability = .18) chance that you will hit two royals, a 6% (probability = .06) chance that you will hit 3 royals, and more than 1% (probability = .015) chance that you will hit 4 royals. If you get really lucky, you could hit 7 royals in 50,000 hands, but this will only happen about t 1 in 10,000 times (i.e. if you were to play 10,000 sets of 50,000 hands, then you would only expect to hit 7 royals in one of these sets).

The take home message here is that royals may happen in clusters. While over the long haul, you will probably hit as many royals as you are expected to, they may occur in small clusters. Don’t let this disturb you, just play in the knowledge that you are expected to win in the long run if you play the full-pay machines with the correct strategy.

Recently, many casinos have introduced multi-play games which allow you to draw one hand and then get multiple (usually three or four) draws. These multi-play machines are great fun, but should be approached cautiously because they allow you to bet 3 or 4 times as much money as a normal machine. You can often play 2000 hands per hour on these machines and can expect to hit a royal in a full weekend of play. Roughly speaking, a four-play game will be about twice as volatile as a normal game because the variability increases with the square root of the number of games played at once (remember the distributions where the 20,000 hand distribution was about twice as wide as the 5,000 hand distribution?). If you only play these machines after you’ve won some money on single-play machines, you may avoid the big losses.

On average, you won’t be making a lot of money playing video poker on a positive expectation game. If you play the most profitable game available (that returns 0.7%), and you play 800 hands per hour (this is pretty fast), you can expect a gain of about $7 per hour if you make no mistakes. This will not make you a sultan. You might hit a good streak and win a few thousand or hit a bad streak and lose a few thousand, but in the long run, this is no way to make a living. With comps thrown in, your hourly rate is more like twenty or thirty dollars, which is not too bad.

If you play 500,000 hands of full-pay deuces-wild in the next ten years - you should earn about $5000. On top of this, you could also get about 30 free vacations where you did not pay for a room (or for food and drink) which might be worth an additional 10 or 15 thousand dollars. If you play a short-pay version (or a full-pay version with only good strategy) - you would be expected to lose about $7500. So if you are going to become a video poker player for the long haul, the knowledge of finding the right machines (and knowing how to play) might save you the difference between these outcomes or $12,500 or more.

Of course, it is possible that full-pay versions may eventually be eliminated. It is hard to predict what will happen, but there is a good chance that that full-pay versions will be around for many years (considering that they have been around for several years and video poker is gaining in popularity). Moreover, you can still get your money’s worth playing the short-pay versions when you start looking at the additional