So how is it that you can play 3181,875,200 hands of video poker, one for every possible unique arrangement of the cards, and still have a 37% chance of not hitting your hand?

The difference arises from how people calculate probabilities for ease/convenience, and how the probabilities should accurately be calculated. To illustrate, the probability of rolling a 6 in one roll of a die is 1/6, or 16.67%. If you get two rolls (or two dice in one roll), most people would say that the probability of rolling a 6 is now 2/6, or 33%. However the true probability is only 30.56%. That is because how you should calculate probability is figure out the likelihood you won’t do it, and then subtract that amount from 1 (giving you the resultant likelihood that you will do it). The probability of not rolling a 6 in two rolls is 5/6*5/6: 25/36 or 69.44%; thus the probability of actually rolling a six is 1 - 0.6944, or 30.56%. If you play backgammon, this should be a familiar percentage.

Carrying this forward, in 6 rolls 'on average' you should have rolled a 6 (6/6 = 100%), however the true probability is only 66.51% (5^6/6^6 or 15,625/46,656: 33.49% that you haven’t rolled a 6, subtracted from 1). Said another way, just over a third of the time, after 6 rolls, you still won’t have rolled a 6 with your die. On a single roll of the dice the true percentage and the commonly quoted percentage aren’t that different. When the number of trials increases, the divergence becomes significant.

In the video poker example, the chances of not getting the hand you want are 311,875,199/311,875,200 every time you play. Every trial multiplies this probability against itself, reducing the probability that you won’t get the hand you want (and thus increasing the chances that you will). Calculating it this way, after your ‘average’ of ~178 years (or 311,875,200 trials), your chances of having hit are only 63.21% (because your chances of not having hit it by then are 36.79%).

Because in theory you could play forever and not hit (though the chances of your doing so become increasingly small as the number of trials expands), you can never say that by X number of trials you are guaranteed to have hit. You can only say how confident you are that you will have hit by a certain number of trials. Fortunately, there’s a (relatively) easy formula for calculating this (which I can share with anyone interested). In our video poker example, it yields the following results:

To be 90% confident that you will have hit, you need 718,119,178 trials, or ~410 years.

To be 95% confident that you will have hit, you need 934,294,591 trials, or ~533 years.

To be 99% confident that you will have hit, you need 1,436,238,355 trials, or over 819 years.

To be 99.9997% confident that you will have hit (the classic ‘six sigma’ confidence level) you need 3,966,085,143 trials, or over 2,262 years.

So what is the ‘ideal’ number of hands to play to ensure that 1) your Eternal Soul is ultimately set free, and 2) you don’t get too many superfluous pokes with the pitchfork? Ultimately there is no ‘right’ answer—it would depend on your particular risk tolerance (ergo the gamble), and probably if you had been experienced the poke with the pitchfork or glimpsed the eternal punishment. Being of the more risk-adverse mindset myself, I think I would endure some extra perforation in order to better guarantee my ultimate release.

Hmm… according to the Book of Revelation, Old Scratch took 1/3 of the host of Heaven with him as part of his original rebellion when he was cast out of Heaven. Perhaps they were all of the people who figured they only had to play Video Poker for 178 years in order to hit their hand?

For anyone still awake, class dismissed .

Brent J. Jensen
R-8007
orbis non sufficit