Different Ways of Looking at Losses for -EV Gamblers
Most of the Articles I write are focused on Advantage Play, News or other aspects of the gaming world, but WizardofVegas.com and WizardofOdds.com seeks to appeal to all gamblers, not just AP's, which is clearly evident by the fact that most of the games analyzed on the Main Site are Negative Expectation games. Let's face it: Not all gamblers are mathematically inclined, but even with that being said, many of them would like to stretch their gambling budget as far as they possibly can before leaving the casino for that night or for the trip. It is with this in mind that I want to look at different ways that a Negative Expectation player may stretch his/her gambling dollar.
What seems like the most obvious solution, simply play the game with the lowest House Edge (often Blackjack) is not going to work for everyone. First of all, Blackjack Rules are deteriorating across the entire country, though mostly on the Las Vegas Strip, and these poor rules affect Advantage Players and Negative Expectation Blackjack Players alike. The most obvious of these detrimental Rule changes is playing at a table upon which Blackjack pays 6:5. One might ask, "Blackjacks don't happen that often, so how much of an impact does this have?"
In order to tackle this question, there are a couple of pieces of information that we want to know: First of all, how likely is it that the player is dealt a Blackjack and not the dealer? The reason it is relevant if the dealer is dealt a Blackjack is because both Player + Dealer getting a Natural will result in a Push regardless of what the Blackjack pays. The second piece of information we need to know is how much the player in question is betting, and the final piece of information is how many hands is the player getting in per hour?
We're going to take a look at a six-deck game of Blackjack to answer how likely it is that the Player alone gets a Blackjack and how likely it is that the Player and Dealer both end up with a Blackjack. There are twenty-four Aces, ninety-six Faces and 312 cards in the shoe, so:
(24/312 * 96/311) + (96/312 * 24/311) = 0.047489488
I prefer to express that as 1 in x, because most people can relate to that, so we simply take one and divide it by the probability expressed as a decimal:
1/0.047489488 = 1 in 21.0573
With that done, we need to look at how many Blackjacks are unaffected by the short pay, which is to say, how often would that Blackjack otherwise be a Push anyway. To do that, we simply determine the probability of a Dealer Blackjack based on the remaining cards and then multiply the two probabilities together:
(23/310 * 95/309) + (95/310 * 23/309) = 0.04562062845
0.04562062845 * .047489488 = 0.00216650028
In other words, a Player Blackjack is only affected by a matching Blackjack by the Dealer on rare occasion, making the probability of the Player getting a winning Blackjack .047489488-.00216650028 = 0.04532298772
Which results in Odds of about:
1/0.04532298772 = 1 in 22.0639 (Rounded)
If we take a look at this Wizard of Odds Page.
Then we can see that the hands per hour in Blackjack relative to how many players there are is as follows:
Imagine that you are playing Blackjack at $10/hand Base Bet, on a 3:2 Blackjack game, a winning Natural would get you a $15 win, however, on a 6:5 game, a winning Natural only results in a $12 win. You are losing $3 per Natural Blackjack, which again, happens when it happens regardless of the skill level of the player. Some people might say, "Well, I'm only losing $3 on these Blackjacks, that's not so bad." I would reply, "I'm glad you're so rich!"
If we take that 1 in 22.0639 probability of getting a Blackjack, and express it as (1/22.0639) and then multiply by 3x the number of hands per hour, then we can get a look at the dollars lost, per hour, of playing the 6:5 game v. the 3:2 game based on how many players are at the table:
In the worst-case scenario, the player is losing nearly three bets worth, per hour, because the player is winning! In the best case scenario, a packed seven players at the table, the player is still losing an additional 70.7% of one bet per hour because of this poor Rule.
How does that compare with the Expected Loss of a $25 Blackjack game in which the Blackjack pays a proper 3:2, and assuming Optimal Strategy? We're going to use six-decks, Dealer Stands, Double on Anything, Resplit to Four Hands, NO RSA, NO HSA, No Surrender, Blackjack Pays 3:2 and we get a House Edge of 0.40312% or .0040312 with Optimal Play. If we change the game to 6:5 Blackjack, we end up with 1.76281% House Edge or .0176281.
If we go back to our 209 hands per hour with one player, then we see betting $25 a hand Base Bet, the player will make a total of $5,225 in bets with an Expected Loss of $21.06. I don't even need to compute the Expected Loss on the 6:5 game, in this instance, because I know that the player is losing MORE MONEY PER HOUR just eating the 6:5 Blackjack pays than he loses on the $25 game! That said, I will go ahead and do it, and for the $2,090 that the $10 player will have in Base Action, he loses $36.84!
How do you like that? The $10 player bets 60% less in Base Bets and loses almost 75% more money! It might be just me, but I don't think that $10 Table is saving that player more money per hour or otherwise. In fact, the $10 player would only even be saving anything per hour in this event:
x * .0176281 = 21.06
PROOF: 1194.68 * .0176281 = 21.06 (Rounded)
In other words, in order to achieve the Expected Loss of the $25 3:2 game, the player in question could only have $1,194.68 in total Base Bets, which is roughly 120 hands. This player would need to play with about three players at the table in order to achieve this, however, the $25 Table might also have more than one player. Either way, these same equations can be used to determine the Expected Loss dependent on the Table Minimum, Rules and Number of Players per table to determine the lowest Expected Loss Per Hour on a given game compared to another.
The point of the matter is: What appears to be the lowest cost option on the surface for Blackjack, or many other games, often isn't. One of the aspects that makes 6:5 Blackjack so successful in these Strip Casinos, in addition to the fact that many players will stupidly tolerate it, is that it makes more money off of a $10 bettor than 3:2 Blackjack makes off of a $25 bettor given this Rule change alone, again, assuming the same number of hands per hour.
In the next two segments of this Article, we are going to first look at House Edges v. hands-per-hour in some other games to determine what games offer the lowest cost to the player both in terms of Expected Loss and Expected Loss Per Hour, as those two things are not created equally. In the final Article, we will take a look at how you might want to go about deciding what game is the best for you to play personally.