I've written many columns in the past several years about how our minds can play tricks with us regarding odd results. We have a tendency to remember what appears to be very rare events and all but forget the routine. The end result is that we begin to think that all that occurs are those strange occurrences. This eventually leads many to believe that a game is either broken (if we are winning) or rigged (if we are losing). The most likely real answer is two-fold. First, we're not correctly remembering what actually occurred and second, most people really aren't aware as to how rare or common some event actually is.
How much does it matter if our memories are a little faulty? In the grand scheme of things, perhaps not much. But, from a math perspective, it can make a great deal of difference. Several months ago, a friend of mine relayed to me his experiences while playing Let It Ride. He claimed that in a five-hand span he was dealt a Four of a Kind, a Full House, a Straight and TWO Three of a Kinds. He must have been able to read the expression on my face as he kept telling me that it "REALLY" happened. I was a bit skeptical.
When I got to my computer I did some computations and discovered that the odds of being dealt those 5 hands in a row (in any order) was about 27 BILLION to 1. Okay, I wouldn't call it impossible, but I wouldn't call it very likely. But, what if his memory was a little flawed? What if he was dealt those 5 hands in a block of 10 hands? What if his memory just 'forgot' about the five losing hands? In this case, the odds would drop to a 'mere' 492 MILLION to 1. At least we're back down into Lotto territory. So, did my friend get these 5 power hands in 5 deals? 10 deals? 20 deals? I have no clue.
This past week, I had the opportunity to do my own research. Family is visiting from out of town, which usually means I'm playing in the casino a bit more often. On one evening, I sat down at a five-play machine. In the first 31 hands, I was dealt 5 Three of a Kinds (on the first 5 cards). I know these numbers to be true not because I sit there and count how many hands I've played, but rather I was able to see the point counter on the machine and specifically took notice of how many hands I had played after being dealt my 5th Trips.
This made me wonder about just how rare is it to be dealt 5 Three of a Kinds in 31 hands. So, when I got home and went to work on the calculation. Before you read any further, I'd like you think about this and come up with how often you think this happens? 1 in 100? 1 in 1000? 1 in a million? This is one of the cases where I think many people would guess rather wrong if they don't know how to do the actual math. Until I did the calculation, I didn't really know what to expect. I knew it wasn't astronomical, but I figured it was a bit more rare than it turned out to be. In the end, the number was 562. The odds of having 5 Trips in 31 hands is in the same ballpark as the odds of being dealt a Flush on the deal. It didn't seem like such an odd occurrence anymore, but at the time, I figured the machine was on fire.
Of course, I didn't draw a single Four of a Kind out of any of these hands. Five times I had five chances to get those Quads and I couldn't hit a single one. So, maybe the machine was actually cold, not hot. How cold was the machine that I couldn't hit a single Four of a Kind from my 5 Trips? In reality, not at all. I had an 80% chance of not hitting any Four of a Kinds from these trips. So had I actually hit one (or more), it would've have been the more rare occurrence.
There are many things to take away from this column. One is that it is hard to rely on anecdotal stories. If you didn't witness the event yourself, you don't even know if the story is true. And, even if you did witness it yourself, we necessary learn a lot because once upon a time something rare happened. We need to look at all the stories everywhere in order to learn what to expect. This is best done by computer programs and computer simulations. From this, we learn that virtually everything that happens when we play in a casino is just part of the normal ups and downs that happen 'randomly'.