Ask the Slot Expert: An unusual video poker experience?

Question: Had a very interesting 7-hour triple-double video poker session (25 cents per hand) yesterday at the local Indian casino. Ended the session with gross winnings of $1,020, and no royals (had one the previous 5-hour session).

I would think that is a bit unusual.

What do you think?

Answer: We can actually calculate how likely it is. I have done it to create graphs that show the distribution of returns after playing a certain number of hands. Don Catlin told me that I could call the process "convolution". This process is pretty convoluted.

Here's how it goes. The outcome of our first hand can be one of only a certain number of outcomes -- royal flush, straight flush, full house, nothing, etc. For Triple Double Bonus, there are 14 possible outcomes. We know the probability of achieving each outcome and we know how much each outcome pays.

If we drew a diagram of what could happen, we have a starting node. Then we have a line to the royal flush node and its 4000-coin payoff and we have a very small probability that we follow that path. Another path takes us to a flush and we have a much higher probability to follow that path. For Hand 1, we have a node for each outcome and the probability of following the path to that outcome.

Let's say we got a royal for Hand 1 (Yippee!!). What are the possible outcomes for Hand 2?

All 14 outcomes are possible and we know the probability of following the path to each outcome. There's a really small probability we follow the path to the second royal. So there's a really, really small probability that we follow the path to a royal on Hand 1 and a royal on Hand 2. There's a much greater probability that the next step will be to a nothing hand.

If there are 14 possible outcomes for each hand, the total number of distinct paths is 14 raised to the number of hands played. We go from 14 to 196 to 2744 to 38,416 to 537,824, and so on. The number of distinct paths gets really big really fast.

Fortunately, we can limit the amount of data we have to deal with. First, we don't care about distinct paths, only how much we've won. We don't care how we ended up on the +10 coins node at the end of Hand 2, for example. It could be either two Jacks or Better, two Two Pair, one of each, or 3-of-a-Kind and nothing. We need only one +10 node at Hand 2. The probability of reaching it is the sum of the probabilities of getting these hands:

Jacks or Better - Jacks or Better
Two Pair - Two Pair
Jacks or Better - Two Pair
Two Pair - Jacks or Better
3-of-a-Kind - Nothing
Nothing - 3-of-a-Kind

I think that's all of the ways a Triple Double Bonus machine could have paid you 10 coins after two hands. Note that on each line you multiply the probabilities together and then add up the products.

We can eliminate many paths by combining all of the paths that lead to the same number of coins paid to us.

The second way to decrease the number of paths to track is to, let's say, cull the herd. One way to do that is to eliminate nodes whose probabilities fall below a threshold level. There's no reason to maintain a royal-royal-royal-royal-... path because the probability of getting a royal on every hand is small.

Another way to cull the herd is to decide how many nodes to pass from one hand to another -- like 1000, 5000 or 10,000 -- and sort the nodes by likelihood and drop the least likely nodes that don't make the cut.

When we're done playing the number of hands we want to play, we can graph the data. The value of each node is the x-axis and the probability of getting to the node is the y-axis. We end up with graphs that look the ones on this page, which is where I got the idea to create graphs for the games that I play.

I made my graphs about 20 years ago, long before I moved to Las Vegas. I didn't make one for Triple Double Bonus. I don't play it because it's too volatile. I did look at a strategy chart for it because it used to be one of the best paytables offered at Suncoast, but I was put off by how complicated the strategy is.

I have played TDB occasionally just to see if I could hit something big. I had won a good amount of free play in some sort of promotion a few years ago, so I decided to see if I could turn it into something big playing quarter TDB. I played for a little while before hitting four aces. That's something big. I was happy.

I watched the credit meter increment. It didn't stop after adding $200. Let me take a closer look at the hand.

I had four aces with a kicker. A grand. Now, that's really something big and an excellent outcome for my free play.

You ended up about $1000 ahead too. But you did it playing nickels. You said you also did not get a royal, but you didn't mention anything about how well you did with quads. I imagine you must have hit quite a few of them.

I agree with you. Your experience was unusual. But you're better qualified to rule on the matter than I am because I don't play TDB. How frequently have you ended up $1000 ahead?

To quote a lyric from a song by Premiata Forneria Marconi out of context, "Sometimes your oyster holds a pearl."

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