I never know if my explanation of how an expected value is calculated is really being understood by my readers. It seems obvious enough to me, but I've been working on casino games for nearly 30 years. In my writings, I've generally tried to explain by using an example which I think explains it clearly enough. I usually use one of a couple of examples that my father originally used in his Expert Video Poker for Las Vegas book. This book starts by using a Three of a Kind and follows up with a 4-Card Straight that is also a Low Pair.
The Three Kind Poker leaves very little to think about. Not even a 3-Card Royal is a match for a Three of a Kind, so it really is just an exercise in explaining how an expected value is calculated. The second example is the type of hand that is faced quite often by Players, but you are still choosing from the lesser of evils. This past week, I was dealt the type of hand that makes you really think about it all.
I was playing what has become my favorite game - Multi-Play Multi-StrikeDouble Double Bonus Poker, which is a game not for the faint hearted. Multi-Strike is a game with high volatility. Double Double is a game with high volatility. The Multi-Play does bring it back to earth a little, but it also raises the cost quite a bit. Multi-Strike is a game with four levels. To begin play you are betting on each hand on each level. So, since there are 4 levels and 5 hands, you are betting 20 units (minimally) for each hand. If you want to play max-coin, you are essentially playing 100 units. This is why most players play for 1 or 2 pennies per unit which translates to $1-$2 per hand. I have seen some play it for 5 cents, but I've never seen anyone play it a higher denomination.
To go on to a higher level, you must achieve at least a Pair of Jacks or Better on the lower level. So, you are dealt an initial hand on Level 1. You choose which cards you want to discard and the result is played 5 times (as per normal 5-play). For each of those final hands that are Jacks or Better, the hand will continue on the next level. So, there is a strong possibility that you will pay for 20 hands and will play only 5 losing hands. If you're dealt a High Pair, then you know that all 5 hands will continue to the next level.
So, what makes this game so fascinating is that how you play a particular hand can impact the probability that you play the higher levels. Each level multiplies the payout. So the second level pays 2x, the 3rd level 4x and the 5th level 8x. Hit a Royal on that top line and you win 8000 units TIMES 8 or 64,000 units. That's $640 in PENNIES if you are playing max-coin. So, back to what happened this past week.
I was on the 3rd level with two hands still in action. I was dealt a 4-Card Royal WITH a Pair of Kings. In Multi-Strike, you have to think twice about throwing away a sure winner as this will allow the hand to continue on with certainty. The expected value calculation is far more complex. While you would not discard a High Pair for a 3-Card Royal in the normal game, it becomes an even worse play in this one because you will lower the probability that the hand will continue.
Now the decision before me was not even close. The High Pair has an expected value of 1.4. The 4-Card Royal has an expected value of 18+. This is in a normal game. While this is offset slightly my the Multi-Strike Variation, nothing is going to make up the 17 coin difference in expected value. My 4-Card Royal is still going to be a paying hand of some type 20 out of 47 possible draws. One of those 20 will be the Royal, which on the 3rd line was worth $160. This time I wasn't choosing between the lesser of two evils. I was choosing between two winning hands. Of course, even with 2 shots at it, I could've pulled a 3 and a 7 off-suit and had absolutely nothing. 27 of 47 possible draws were going to leave me not only winning nothing but with no hand on the 4th level. To break it down, I had a 1 in 3 chance of losing both hands. I had just under a 50% chance of winning one and about an 18% chance of having both hands be a winner. Many of these 'winning' hands would be only a High Pair.
But when we play according to the math, we're playing for the long run. A 4-Card Royal is a rare hand. A large percentage of the time, it will be a Flush, a Straight or a High Pair as well. When it is none of these, it is a very easy decision. When it IS one of these hands, the decision is a bit harder, but just as obvious. You need to play the math. If you don't get your fair share of Royals, you're not going to get the theoretical payback out of the machine. Royals are very rarely dealt to Players. For every Royal you are dealt, you'll be dealt 234 4-Card Royals! You have to play them correctly.
I went for it. The first of the 2 hands was a Flush. A nice payout on the third line. I knew the first hand had no relationship to the second, but a small part of me figured my odds of hitting that Royal on the second hand went down with the winning hand on the first one. I pumped my fist in the air as the second hand completed the Royal! I had a 2 in 47 chance (as I had 2 hands going) and a little luck never hurts.