A Simple Coin Toss

18 November 2004

By Lenny Frome

Readers of this column are well aware that Video Poker is essentially a game which is governed exclusively by probabilties. Given a deck of cards, given a version of poker to be played and given a computer, the stats of the game are determined.

Notice that we determine the statistics of the game, meaning the average number of the kinds of hands dealt out, the Expected Values of all of these types assuming the player is expert and the payback one would expect over a large sample of games played.

In Video Poker, as in most forms of gaming, we can never predict an outcome or result except as a probability , or as odds for/ against an event happening. Understanding of just what these odds imply is very important to a player.

Take a simple coin-toss, with heads or tails as the 2 possible outcomes of a trial, each with a 50% chance. One might think that if tossed 20 times, it would be very likely that 10 heads and 10 tails would occur. In fact, that particular split would be quite unlikely, the true odds being between 4-1 and 5-1, since a 10-10 split is only going to occur in 18% of our 20-flip trials. OK, so well use a 40-flip trial, to increase the chance of a 20-20 even-up split. Worse than before, the odds versus a 20-20 split are nearly 7-1 because it will occur in only 13% of our 40-flip trials.

This kind of perversity is active in our Video Pokers as well. We're always searching for answers to questions like, "How often can an expert player expect to get Four-of-a-Kind?"

Our stats tell us that we can expect Fours once in 420 hands on average. Of course, we knew that a coin comes up heads once in 2 flips on the average, so we are not going to jump to the conclusion we can count heavily on seeing a Fours in 420 trials. As a matter of fact, our chances can be tabulated as follows:

Hands Played	Exactly
Hands Played	0	1	2	3
420	37%	37%	18%	6%
840	14%	28%	28%	19%
126	5%	15%	23%	22%
1680	2%	8%	16%	21%

is can be interpreted to mean that 37% of the time, we will not get a Fours in one cycle (420 hands) but there's a 26% chance of getting more than one Fours. If we play 2 cycles (840 hands) there's a 42% chance we get less than the 2 Fours we expect but a 30% chance of getting even more than two. If we play 3 cycles (1260 hands) there is a 43% chance of not getting 3 Fours and a 35% chance of getting more than 3. In 4 cycles there's a 47% chance of getting under the average 4, but a 32% chance of getting even more.

In general, the longer we play, the less likely it becomes to hit the average value exactly.

Lenny Frome

Lenny Frome spent 40 years in the aerospace engineering business before moving to Las Vegas. During the ensuing 10 years, he became one of the most prolific gaming authors, having written 8 books, countless tip sheets and nearly 1000 articles for a variety of gaming magazines.

Lenny's ground breaking work in the area of Video Poker earned him the title of the 'King of Video Poker'. He also wrote on a variety of other gaming topics including Spanish 21, Let It Ride, Keno and others.

Besides being an author and columnist, Lenny was the premier Gaming Consultant at the time of his passing in 1998. He helped develop paytables for Let It Ride and Three Card Poker, and consulted on literally hundreds of other gaming projects.

His son, Elliot, now follows in his footsteps, as a gaming author, analyst and consultant. Their website, and a complete catalog of all their products can be found at www.vpheaven.com. Feel free to drop Elliot an e-mail at compuflyers@prodigy.net.

Lenny Frome Websites:

www.vpheaven.com