by Mike Aponte on August 3, 2010
In the movie 21, Professor Mickey Rosa poses an interesting puzzle to his class – The Monty Hall Problem. Imagine you’re on a game show and the host presents to you three doors. Behind two of the doors are goats. Behind one of the doors is a car. Choose the correct door and you win the car. Let’s say you choose Door #1. The host, who knows what’s behind all the doors, opens Door #3 and reveals a goat. The host then offers you the choice of staying with Door #1 or switching to Door #2. Should you switch doors at this point? To answer this question you must first determine the probability of choosing the car and the probability of choosing a goat.
At first blush it may seem that it’s basically a coin flip. After all there are two doors left, one of which will reveal the car. Either door would be a 50/50 proposition, right? Well, as counter intuitive as it may seem, if you switch doors you would double your odds of winning. Many people emphatically believe that after Door #3 is eliminated, the odds of wining and losing are equal. This would be true if there were only 2 doors to choose from at the start of the game. But, the game began with three doors which means that when you selected Door #1, you had a 1 in 3 chance of winning the car and a 2 in 3 chance of getting a goat. If you stick with your original pick of Door #1, those odds of winning and losing remain the same.
However, if you choose to switch doors, your odds of winning increase dramatically. The key to this problem is that the host knows where the car is and will only open a losing door. If you switch doors, you will win if the car is behind Door #2 or Door #3 because the host will always open a door that has a goat behind it. You essentially have two shots at winning. By switching, you would only lose if the car happens to be behind Door #1.
In summary, if you stay with Door #1 you have a straight 1 out 3 chance of winning the car and a 2 out 3 chance of claiming a goat. If you switch doors you will win if the car is behind Door #2 (1/3) or Door #3 (1/3). Adding these two probabilities yields a 2/3 chance of winning and a 1/3 chance of losing. The Monty Hall Problem illustrates an important principle of card counting. Card counters use information gained from the cards as they are dealt to update the constantly changing odds of the game. In order to earn a long term profit, a good card counter must capitalize on this by betting proportionally larger amounts relative to player advantage – in the same way Ben Campbell correctly chose to switch doors after determining that it would increase his odds of winning.
by Mike Aponte on April 22, 2010
Many gamblers are intrigued by betting systems based on the amount wagered and the outcome of the previous hand. These systems, which are touted in some books and all over the internet, are betting progressions – the most popular of which is the Martingale. With the Martingale system you start off with your base bet, and if you lose, you double your wager. You continue to double up your bet with each loss until you win, at which point you return to your base unit. If your hand pushes with the dealer, you wager the same amount. This approach produces a high percentage of small profit sessions.
This system may seem like a fool proof way to beat the house. After all, the odds of losing 7 or 8 hands in a row are slim. As long as you finish a session after a winning hand, you will walk away a winner. But, all progression bettors learn the hard way that sooner or later you will get wiped out. Eventually you will lose 7, 8, even 10 or more consecutive hands and your session will end with a devastating loss. When a progression runs into an inevitable unlucky streak, the result is financial ruin. For example, if a Martingale bettor with a $10 base unit loses 10 straight hands, he would drop $10,240. When you’re doubling your bet everytime you lose, it doesn’t long for a small base bet to become a huge bet you can’t afford to lose. If you play long enough, at some point you will be hit with a painful losing streak. I know this all too well, having once lost 14 straight hands in Atlantic City. If I had been utilizing a betting progression, the result would have been disastrous.
Betting progressions could work in a hypothetical world. If you had an infinite bankroll and played at a casino which offered an infinite table maximum, you would be able to whether the storm of any losing streak, without fear of going broke or being constrained by a table max. In the real world however, gamblers play with a limited bankroll and table max’s are far from infinite. House edge is what rules. Simply stated – whoever has the advantage will come out on top. All casino games have a built-in edge for the house which is immune to betting progressions. Due to their simplicty and often positive, short term results, betting progressions may be appealing to unsuspecting gamblers, but unlike card counting and other forms of advantage play, they do not hold up to the unforgiving laws of probability.
by Mike Aponte on February 3, 2010
At most casinos the dealer must “burn” the first card of a shoe by placing it directly in the discard tray before any cards are dealt. In Atlantic City the 
burn card is usually shown before it’s taken out of play, but what about when the burn card isn’t revealed? This seems to cause angst for some blackjack players (“that could have been my ace”), but what impact does it really have on the game? For the vast majority of blackjack players who aren’t keeping track of the count, it has no effect whatsoever. From a card counting perspective, when the dealer burns a card it’s equivalent to moving the cut card up by one. For example, if the cut card is placed at a deck and a half (78 cards), the burn card effectively moves the cut card to 79 cards. There’s simply one more unknown card. For both card counters and the average gambler, the impact is insignificant at best.
If burning the first card doesn’t serve as an effective measure against card counting, why do casinos employ this practice? The purpose of burning a card is to protect against the steering of the top card. If a card is exposed when the cards are presented to players to cut, the exposed card can be cut into play. Steering an ace to your self is worth a whopping 51% in expected win. Cutting a ten to self is worth a not too shabby 14%. Card steering is completely legal if the card information is obtained due to the dealer unknowinlgy exposing a card. The illegal acquisition of card information is what casinos are really concerned about. Cheaters can mark tens and aces, making them recognizable when they end up at the top of the shoe. There’s also the possibility of collusion with the dealer who either flashes the top card to fellow accomplices, or peaks at it and then passes on the information. The bottom line is, even if you’re counting there’s no need to fret about an unseen burn card. Essentially, all the dealer is doing is moving the cut card up by one.
by Mike Aponte on January 20, 2010
Professional gambler has always been a moniker that makes me twinge. As a card counter, I have never considered myself a gambler. Investor is a much more fitting title for a professional card counter. Some may say that this is an insignificant matter of semantics. After all, both gamblers & investors put their money on the line with the goal of making a profit. However, there is a big difference between the approaches & outcomes of a gambler as compared to an investor.
These are the traits of a gambler:
1) A gambler does little or no research & preparation before taking on risk.
2) A gambler hopes to win despite unfavorable odds.
3) A gambler acts based on hunches, misinformation & unproven systems.
4) A gambler is affected by the emotions of greed & fear.
5) A gambler’s motivation is largely driven by thrill seeking & entertainment.
6) A gambler loses.
Conversely, these are the traits of an investor:
1) An investor completes thorough research & preparation before taking on risk.
2) An investor knows he/she has a high probability of winning (making money) because the odds are in his/her favor.
3) An investor utilizes a rational & proven model or system.
4) An investor does not allow emotions to influence his/her decisions.
5) An investor’s motivation is not risk seeking & entertainment.
6) An investor wins.
These characteristics hold true not only in blackjack, but in just about any potential gamble/investment, whether it’s the stock market, poker, real estate or buying a fast food franchise. Gamblers tend to have a narrow, short-term view of risk versus reward, focusing primarily on the upside. On the other hand, investors see the big picture & factor in risk when evaluating the long-term prospect of an investment. Of course even the savviest investment has an element of chance. But, if you have the right mindset & implement a rational, proven approach, you will maximize your odds of success.
by Mike Aponte on April 17, 2009
Most blackjack players are oblivious to whether the dealer stands or hits on soft 17. The dealer stand on soft 17 rule has always been the standard, but a growing number of casinos now require the dealer to hit on soft 17 (hands like Ace, 6). Not only do most players fail to recognize the difference, some believe the hit soft 17 rule works in their favor since the dealer will bust more often. The dealer will bust more often, but the dealer also has the opportunity to improve on a weak total of 17. The reality is the dealer will improve his hand often enough that it will more than offset the increase in dealer busts. It’s not a good feeling when you have 20 and the dealer hits on a soft 17 and pulls 21 to wipe out the table.
Below are the dealer outcome probabilities for a 6-deck shoe for the stand soft 17 rule versus the hit soft 17 rule.*
18 19 20 21 Bust
Stand Soft 17 14.62% 14.04 18.85% 7.65% 29.60%
Hit Soft 17 14.82% 14.24 19.06% 7.86% 30.00%
As you can see, when the dealer hits soft 17 rule is in effect there is a greater probability the dealer will draw to 18, 19, 20, or 21. The net result of the hit soft 17 rule is that it adds .22% to the house edge. All else being equal your odds are better when the dealer stands on soft 17.
* Probabilities rounded to the nearest one hundredth of a percent. With the dealer stands on soft 17 rule, the probability of the dealer drawing to 17 is 15.25%. With the dealer hits soft 17 rule, the probability of the dealer drawing to a hard 17 is 14.02%.
by Mike Aponte on April 10, 2009
As the dealer lays down your second card, a smile grows across your face. You just got blackjack. A moment later your smile fades. The dealer has an ace showing. She calls out for insurance bets. Before the dealer checks the hole card, she makes eye contact with you, awaiting your decision. The player next to you advises, “I would take it. Even money is the smart move. ” The dealer agrees. “Always take insurance when you have blackjack. It’s a sure thing.”
To take insurance or not take insurance. It’s a decision that becomes tougher for players when they have a good hand, particularly blackjack. The more money on the line, the more likely players are to take insurance. This is where players go wrong. Insurance is not a wager you should take to “protect” your original bet. It is striclty a side bet on whether or not the dealer has a ten or face card in the hole. Players can insure their bets by placing an amount up to half their wager in the insurance area on the table. If the dealer does have a ten or face card in the hole, the bet pays off 2 to 1.
Let’s break down the expected win of insurance for a 6-deck shoe. When the dealer shows an ace, the odds she will have blackjack are 96/311. There are 96 ten-value cards in a 6-deck shoe and there are 312 total cards in a 6-deck shoe (52 x 6). Not counting the ace showing there are 311 cards remaining. There is a 96/311 chance that the dealer will flip over a ten or face card and pay off insurance bets 2 to 1. There is a 215/311 chance that the dealer will not have blackjack and all insurance bets will lose.
Based on a $100 bet, here is how the expected win works out.
(96/311) x ($200) + (215/311) x (-$100) = $(19200/311) -
$(21500/311)
$(19200/311) – $(21500/311) = -$(2300/311)
= -$7.395
Over the long run, for every $100 you wager on insurance you can expect to lose $7.395 . The insurance bet has a 7.395% house edge. To put this in perspective, the house edge on insurance is more than 14 times the edge versus a basic strategy player, playing at a .5% disadvantage. This is why one of the cardinal rules of basic strategy is never take insurance. So the next time someone tells you to take even money, remember, the only sure thing about insurance is that it will cost you a pretty penny over the long run.
Blackjack: What are the Odds?
by Mike Aponte on April 2, 2009
Blackjack is the name of the game. So what are the odds of getting dealt a blackjack? The probability depends on how many decks are played. Of course the essential ingredients to blackjack are an ace and a ten-value card (10, jack, queen, or king). In a single deck game there are 4 aces and 16 ten-value cards. You can get blackjack in one of two orders:
1) A ten-value card followed by an ace
2) An ace followed by a ten-value card
If we add the probabilities of these two events, this will give us the odds of getting blackjack.
The odds of getting a ten-value as your first card is 16/52. 16 ten-value cards divided by the total number of cards in 1 deck. The fraction 16/52 reduces to 4/13. The odds of getting an ace as your second card are 4/51. 4 aces divided by 51 cards remaining. Multiplying 4/13 by 4/51 gives us the odds of getting a ten-value card followed by an ace.
1) 4/13 x 4/51 = 16/663
Looking at the second scenario, the odds of getting an ace as your first card is 4/52 (4 aces divided by 52 cards) which reduces to 1/13. The odds of getting a ten-value as the second card are 16/51. 16 ten-value cards divided by 51 cards remaining.
2) 1/13 x 16/51 = 16/663
Adding the probabilities of the two different orders in which you can get blackjack (16/663 + 16/663) yields the total probability of getting blackjack.
16/663 + 16/663 = 32/663 = 4.827 %
For a 6-deck shoe, the same principles apply but the number of cards changes. In 6 decks, there are 24 aces, 96 ten-value cards, and 312 total cards.
1) Ten-value cards / Cards Remaining x Aces / Cards Remaining
(16 x 6) / (52 x 6) x 24 / (52 x 6) – 1
96 / 312 x 24/311 = 288/12,129
2) Aces / Cards Remaining x Ten-value cards / Cards Remaining
3/39 x 96311 = 288/12,129
The sum of the the two probabilities is 4.749% which means that you’ll get blackjack once out of every 21 hands.
by Mike Aponte on March 26, 2009
Brian is going with the underdogs for his Thursday and Friday NCAA Tournament picks. All Vegas lines are based on line at Caesars Palace on 3/25 as noted on Vegas.com.
1. Connecticut -7.5 vs Purdue. I’ll take the points and go with Purdue here. I do think UConn wins this but it will be a little tighter than the spread. Purdue put up some solid first round wins while UConn coasted against lesser competition. And I’ve got to wonder if the recently announced potential NCAA violations might weigh on the minds of UConn and Coach Jim Calhoun.
2. Duke -2.5 vs Villanova. I like the underdog again and have this game as a virtual “pick’em”. Both teams got a big home court advantage in the early rounds and were able to get through one tough game (Villanova vs American; Duke vs Texas). Villanova matches up pretty well with Duke and has been to the Sweet 16 several times. I think this will give them enough of an edge for me to take Villanova and the points here.
3. North Carolina -8.5 vs Gonzaga. UNC has looked good in the tournament while Gonzaga has had two tough games. The big question is still the state of Ty Lawson’s big toe. UNC should win this game but I think it will be a little closer than the spread. Take Gonzaga and the points.
4. Michigan State -2 vs Kansas. This is a rematch of a game played earlier this year in East Lansing, won easily by Michigan State. I believe the Jayhawks get revenge here. Bill Self has an incredible record in “revenge” games where he’s had a chance to pay back a team for a loss earlier in the same season. The team has grown since then and the numbers show Kansas may even win this outright by a point or two. Go with Kansas here.
by Mike Aponte on March 25, 2009
Brian’s picks went 0-2 on Sunday, making his record so far in the NCAA tournament a combined 2-7. So what does Brian’s 2-7 win-loss record tell us about his handicapping system? Very little. It’s not possible to determine the long term advantage of Brian’s system based on a tiny sample of 9 picks. Evaluating a handicapping system based on 9 picks is like trying to assess if a coin is biased after only 9 flips. Even in games or ventures that present the opportunity for skilled and knowledgeable players/investors to gain an edge, natural variability has much to do with what happens in the short and mid-term. Given a large enough sample size however, it is possible to determine if a system or hypothesis is valid.
Blackjack has an inherent advantage over sports handicapping and financial trading in that the system for winning at blackjack is black and white and has been proven for more than 40 years. Back in 1959, Professor Ed Thorp had a hypothesis that blackjack could be beat. Thorp formulated and tested his system using the supercomputers at MIT. Computer simulations played out millions of blackjack hands, and confirmed that basic strategy is the optimal playing strategy and that card counting yields a long term advantage for players. The validity and effectiveness of card counting has been irrefutably proven; when most blackjack players fall short it is due to a lack of knowledge or a lack in skill in applying the system.
For those who wish to make sports handicapping or financial trading a profession, a model must be built and then tested to determine if it will make money. The keys to building a successful model are developing a hypothesis, testing the hypothesis by crunching numbers using an ample and reliable database, defining a model, and back-testing the model by applying it to historical data to see if it’s profitable. The goal is to arrive at the predictors which forecast future outcomes such as final scores and market movements. Even if back-testing the model indicates profitability there is a possibility the model may no longer be as profitable or viable at all. Since the back-testing is based on past data, Vegas line makers may now take the predictive model into account or the stock market may have adjusted to the market bias. This is an issue that does not apply to blackjack since the variables (house rules and conditions) are very easy to account for when determining the value of a a particular game. In future posts we will explore in more detail how to build predictive models.
by Mike Aponte on March 21, 2009
Brian’s picks went 1-2 on Saturday. The following are Brian’s picks for Sunday.
1. Syracuse -1.5 vs Arizona State. I show this game as essentially a pick ‘em with a very slight edge to Arizona State. The computer models continue to be unimpressed with Syracuse despite their easy first round win. Arizona State showed nicely in the first round for what may have been an underrated Pac-10 conference. The bettors seem to be with me here as this one opened as Syracuse -2.5.
2. Xavier -4 vs Wisconsin. I really like Wisconsin in this one. I think Xavier wins…but this could another one of those overtime classics. The Badger nation will fill up the auditorium in Boise and they may even sneak this one out.
