We’ve all done it. We’ve all sat at a machine and questioned its “fairness.” When the machine goes into one of those cold streaks, we start to wonder if it is rigged.
Of course, we never do this when the machine is on fire. It’s amazing how even as adults, when we are losing, the other side must be cheating and when we are winning, everything is good in the world.
Recently, while playing a 5-play machine, three rather “odd” events happened in a short period of time. I only played for an hour, so while these events didn’t happen one right after the other, they did occur in this relatively small sample size.
The first was that I was dealt a 3-Card Straight Flush on the deal. I subsequently hit three of them for Flushes. Next, I was dealt a 2-Card Royal and two of them drew to Flushes. Lastly, I held a 3-Card Straight Flush again and drew one Straight Flush.
As I sit here to write this column, I don’t absolutely know the answer to the questions I’m about to ask. What are the probabilities of these events happening and just how rare are they? And, which of these events was the most “odd?” I am only looking at the final result given the deal, not the actual deal itself.
Let’s take a look at them one at a time. For event one, I was dealt a 3-Card Straight Flush and the resulting draw left me with three flushes. There are 10 remaining cards of my suit, so there are 45 ways I can pull two cards from this suit on the draw (Some of these will create a Straight Flush, but for simplicity, I’m counting these in my results – as I don’t completely remember what type of 3-Card Straight Flush it was (inside, double inside, etc.).
There are 1,081 different 2-card draws. So 45/1,081 = 4.16%. But I got three of these, so we need to multiply this by itself 3 times. This gives us 0.0072% or 1 in 13,862.
However, this is not our final answer. I had five shots at getting this hand. I got it three times. There are 10 different ways I could get three hands from five chances (mathematically 5 choose 3). I need to multiply this result by 10, which brings us to our final answer of 0.0721% or 1 in 1,386. All of a sudden, this doesn’t seem all that odd.
Next up is my 2-Card Royal. Part of the reason we keep a 2-Card Royal is not only to hit the Royal, but we have our shots to hit a Flush. I have to admit I was a bit shocked when I came up with two of them on the same draw!
Again, for simplicity, I’m ignoring any combinations that would have resulted in a Royal or Straight Flush. There are 165 ways I can draw three of my suit out of 16,215 possible 3-card draws. This is just over 1% probability.
This happened twice so I need to multiply this by itself. This gives us about 0.01% or 1 in 10,000. But as in the prior example, I had five shots to get these two hands. It turns out 5 choose 2 is the same as 5 choose 3 and I need to multiply this by 10 to get to our final result. When I do this, I get 0.1035% or 1 in 966. How do you like that? This is actually a bit more likely than event number one?!
Next up was my Straight Flush. This one I remember was a 3-Card Double Inside Straight Flush draw. So, there was only one way to hit the Straight Flush out of 1,081 possible draws. Again, though, I had five chances to hit the Straight Flush. We multiply our result by 5 and we get 0.4625% or 1 in 216. It really wasn’t much of a surprise to me that this was the most common of the three events. I recall hitting a Straight Flush before from a 3-Card Straight Flush. I did not recall either of the other two events ever happening – but that doesn’t mean they hadn’t happened before.
It should be noted that event was actually rarer than even drawing to the Straight Flush, even if I had only one shot at it. But, none of these events was truly all that rare. One in 1300? The probability of being dealt a Full House on the deal is about 1 in 695. That happened also in this session. Of course, something that occurs on the deal will seem more normal to us because it happens overall more often.
The odds of getting my three Flushes from a 3-Card Straight Flush was 1 in 1386. But what are my odds of playing a 3-Card Straight Flush?
From a quick calculation, I got about 2.25% of our hands or 1 in 45 hands. So, if 1 in 45 hands is a playable 3-Card Straight Flush and 1 in 1386 of these will have three turn into Flushes, overall we are talking about an event that happens once in 63,000-plus hands. I’m likely to hit a Royal Flush before the next time this odd event happens.
But hitting a Royal doesn’t mean the machine is broken, nor does hitting a once in 63,000 hand event mean it is broken. Somebody out there has been dealt a Royal Flush on the deal and this is a 1 in 600,000-plus hand shot. I just haven’t been the person it has happened to!