I sometimes go to the casino with a pal of mine and we combine our cash, play the same machine and split the profits, if any. When we go for a while without hitting a 'good' hand (usually a 4K), we call that a 'dry' spell and hope when we finally hit one that the dry spell is over. That's a normal human reaction; we hope that now a 4K has shown up, others will follow almost immediately. It seldom happens, but it's nice to have hope when engaging these soulless, cold, uncaring devices in the battle for your bucks.

We have all felt at one time or another that we were 'due' for a Royal or a Straight Flush or some other such rare event, but most of the time we're wrong and then - just as we've abandoned all hope - here it comes! For a period in late 1998, I played about 186,000 hands of Video Poker without getting a Royal Flush. Now the frequency of a Royal varies from game-to-game, but for the ones I play most often, a Royal will appear once about every 40,000 hands. So, during those 186,000 hands, I 'expected' to receive at least 4 Royals, yet got none. Is this unusual, or can you expect the same thing to happen to you? Like so much of this, the answer lies in the mathematics of the situation.

Let me break the math down to a simple event: the flipping of a coin. We 'expect' to see heads half the time and tails the other half, but we all know it isn't that precise. Can we have two heads in a row? Sure. What about five heads in a row; can that ever happen? Yes, it can, but not very often. To figure that probability, we multiply the probability of heads ( 50% or .5) times the number of events. Thus, the probability of five heads in a row is .5 X .5 X .5 X .5 X .5 = .03125 which is a 1 in 32 event. So, if the flipping of a coin five times is one 'trial', you'll need about 32 such trials to get 5 heads in a row. Now understand that it may happen on your first trial, or it may not happen until trial # 50; but it will eventually happen.

It's the same with a Royal. If we 'expect' to see a Royal once every 40,000 hands, it may happen right away, or it may take much longer than 40,000 hands. The real question is: What's a reasonable 'waiting' time? Was my 186,000-hand dry spell the norm? Here we need to introduce another mathematical concept called 'standard deviation'. This is simply a way of expressing the differences we can experience between 'expectation' (half heads, half tails) and reality (five heads in a row). Standard deviation (SD) is calculated in matters of degree. One SD will cover what happens about two-thirds of the time, two SDs covers what will happen 95% of the time and three SDs covers 99.7%. Thus, if you experience a 2+ SD event, it's basically a 1 in 20 happening; you'll see what that means in just a minute. I had an expectation of 4+ Royals from 186,000 hands but received none. Just what is the standard deviation for such an event? To calculate the SD, we take the square root of 186,000 which is 431.3 and the square root of the probability of a Royal (1 divided by 40,000 = .000025 and the square root of that is .005, so 431.3 X .005 = 2.16. This means that in 186,000 hands of play, we can expect 4 Royals (since we can't win a 'half-Royal'), but 66% of the time we'll receive 4, plus or minus 2. Therefore, a result somewhere between 2 and 6 Royals in 186,000 hands of play isn't at all unusual. A two-SD result would be between zero and eight Royals and that was my experience. Hitting zero Royals in 186,000 hands is rare, but not so rare that it would make me think the games were dishonest or that I was playing incorrectly. Now, if I had played 400,000 hands where I'd expect 10 Royals and hit none, I'd be worried. The math shows us that: The square root of 400,000 = 632 X .005 = 3 Royals is one SD. No Royals received in a 400,000 hand trial is a 3+ SD event and that's as rare as it comes.

Just so you don't worry about your ol' pal here, after hitting the Royal which ended the 'dry spell', I went on to hit two more for a total of 3 in 52,000 hands of play. That, in itself, is another rare event. The square root of 52,000 is 228 and if we multiply that by .005, we get 1 for the SD. Expectation is to get one Royal in 52,000 hands of play; I got 3. So, my result there was a 2SD event. But it was the other way, so naturally it's easier to take.

What about a 4K, that wonderful hand that keeps us alive at Jacks or Double Bonus games...how long a dry spell can we see here? I remember playing Jacks or Better for 7 hours once without hitting a 4K; that represents about 4900 hands. The probability of a 4K is once every 425 hands, so my expectation is 4900 divided by 425 = 11. The probability is 1 divided by 425 = .00235 and the square root of that is .049. The square root of 4900 is 70, so the SD is 70 X .049 = 3.5. That was a 3SD event, something which has a 1 in 300 chance of happening. And you guys think I'm lucky! I will say, for the record, that I'm sure the machines on which that happened are honest and I haven't had such a dry spell since.

See you here next time.