A cheerful Watson strode into the study, only to be filled with dismay. The great detective was slumped back in his armchair, eyes glazed over, staring straight at, but somehow not through, the windows ahead.
“Holmes! What is it?”
“The solution,” Holmes droned. “The solution.”
Watson’s face dropped. His shoulders sagged.
“Oh, my dear man,” he said mournfully. “How many times have I warned you … ”
Holmes leaped to his feet, eyes flashing, suddenly alert.
“You mistake me, Watson. You and I are going on a little trip.”
He strode to his desk, and lifted an envelope and unfolded the contents.
“Read this,” he commanded.
“Why, it’s from America,” the doctor exclaimed. “Are you to journey to America?”
“Read, Watson. Read.”
“I will, Holmes. I will.
“ ‘My dear Mr. Holmes, your fame has spread across the ocean as a man who can see around the corners, untangle knots and peer through any mystery that might present itself. My problem is quite a common one, I’m afraid, and yet I find myself unable to penetrate its depths.
“ ‘For the last decade or more, I have traveled to casinos as part of my entertainment. There have been times I have won money. More often, I leave a small sum behind. That does not bother me. I understand that the games have odds designed to part me from my money, and I have learned to celebrate the winning sessions when they come, and enjoy the entertainment while minimizing my risk when I lose. I do not wager more than I can afford.
“ ‘However, I find myself wondering if there might be some games that give me a better chance to win than others. I play a little of everything, risking a little at the roulette wheel, spinning the reels on the slot machines, maybe taking a flutter on blackjack or craps.
“ ‘I have heard that it is possible to win more often by counting cards at blackjack or learning strategies on certain video poker games, but I am afraid that I do not have the time or patience for such things. I don’t fancy myself becoming an expert by any means, but if there is a way to give myself a better chance to win, or at least have a little more of my money remaining at the end of the day.
“ ‘Mr. Holmes, do you have a solution?’ ”
Watson looked puzzled. He’d been amazed to watch Holmes penetrate murders, stolen gemstones, concealed identities and international plots. But this …
“And do you, Holmes? Do you have a solution?”
“A 2 percent solution, Watson. Come. Let us go.”
It was not to America that the detective guided the doctor. Instead, he chose a nearby casino with an American-style mix of games.
“Do you know why casinos make money, Watson?”
“I suppose it’s that the games are set up so that the house side wins more often than the player side.”
“Nearly, Watson, nearly. Not quite correct, but quite good.”
Watson looked perplexed and slightly annoyed, but waited for the explanation.
“I’ll show you what I mean,” Holmes said. “Let’s walk over to that baccarat table.”
The table was just modestly busy, perfect for Holmes to point out the clues.
“You see that only one hand of cards is being dealt, instead of each bettor receiving his own cards? Instead of wagering on his own hand, the bettor may wager on which side wins, player or banker.”
Watson thought hard.
“Then how is it possible for the house to make money, Holmes? After all, one side or the other must win.”
“Or it could be a tie, Watson. But on ties, those who wager on either player or banker get to keep their wagers.”
“Then how … ”
Holmes stared, stone-faced, then let a tiny smile crack. Finally, the great detective guffawed heartily.
“Oh, Watson. Permit me this small laugh at your expense. You see, I have information that you do not. The banker side wins more often than the player side. Therefore, the house has an advantage on bets on player. BUT … the house collects a commission on winnings when the banker side wins.”
“I see, Holmes, now that you’ve explained it. The house can lose more than half of the time, and still win.”
“That is correct. In the case of baccarat, the house has a 1.09 percent advantage on banker wagers, and 1.24 percent on player. To use American standards, for every $100 wagered on banker, the house expects to keep $1.09. For every $100 wagered on player, the house expects to keep $1.24.
“Both advantages to the house, you will consider, are less than 2 percent.”
Watson’s eyes widened and his jaw dropped. It was as if a new day had dawned.
“And THAT,” he said, “is the 2 percent solution.”
“Exactly, Watson. Exactly. I will tell my American correspondent that without spending a great deal of time and effort learning to count cards in blackjack, learn to recognize the best video poker games and applying the most advantageous strategies to them, or learning to control the dice at craps, he cannot hope to win consistently. However, he can win a little more often, keep some of his money more of the time, if he will limit his wagering to the games that give the lowest advantage to the house.
“Further, I shall draw that line at 2 percent. If a game has a house advantage of 2 percent or less, it goes on a list of games my correspondent might consider playing. If the house advantage is more than 2 percent, it goes on a list of games he might consider avoiding. The choice, of course, is his.”
Watson’s curiosity was raised.
“What reasons might he have for playing the games on the list of those to avoid?”
“Oh, there are many reasons, Watson. He told us in his letter he travels to casinos as part of his entertainment. Perhaps he simply enjoys some games that are more advantageous to the house. Perhaps he finds them, for lack of a better word, ‘fun.’”
Holmes raised an eyebrow at his own statement, then continued.
“Look around you, Watson. What do you see most?”
“Why, slot machines, Holmes. There might be a few hundred chairs at gaming tables, but there are thousands of slot machines. Are they part of the 2 percent solution?”
“No, Watson, they are not. The house advantages are quite high. On low-denomination machines, such as the American one-cent and two-cent games, house advantages tend to exceed 10 percent. Even on higher denomination games, such as American dollars, the house advantages usually are between 4 and 6 percent, depending on the competitive situation of the host casino. Additionally, we must consider that play on slot machines is very rapid. A slot machine enthusiast can make many more wagers in an hour than can a table games player. So no, Watson, slot machines are not part of the 2 percent solution.”
“Then tell me, Holmes, why are they getting most of the play?”
“People who wager in casinos find them fun. There are games within a game on many of them --- there is more here than meets the eye. And slot machines offer the chance at a big jackpot of thousands, sometimes even millions of dollars. Whether they win or not, players enjoy the anticipation as they wait for the jackpot mystery to unfold. If my correspondent truly enjoys that kind of entertainment, as many do, the 2 percent solution is not for him.”
With that, Holmes whirled and paced quickly to a blackjack table.
“This, Watson, can be an element of the 2 percent solution, but for many it is not. It is one of the few games in which it is possible for players to gain a mathematical edge on the house.”
“The counting of cards mentioned by your correspondent, Holmes? How does that help?”
“Unlike the mathematics in most games, those in blackjack change with each turn of the cards. A card removed from the deck changes the composition of the deck remaining to be played. A greater concentration of high cards increases the percentage of two-card 21s --- blackjacks --- that are dealt. That favors the player, so counters of cards raise their wagers in such situations.”
“I understand, Holmes. But the American said he had neither the time nor patience for counting cards. Can this game still be part of the 2 percent solution?”
“It can, Watson, if he will take the time to learn what is called ‘basic strategy,’ and learn the most advantageous times to take another card, or to stand on what he has, or to split pairs of like cards to make two hands, or to ‘double down’ --- that is, to double his bet and taking just one more card. If he learns that strategy, he reduces the house advantage to just about half a percent, or a few tenths more or less depending on house rules.”
“The rules differ from house to house, Holmes?”
“Indeed they do, Watson, and even sometimes from table to table. Still, if my correspondent will take the time to learn basic strategy, blackjack is well within the 2 percent solution, or even a 1 percent solution. If he will not, then blackjack becomes a borderline game. An average player, one who knows a little basic strategy but misses some fine points, faces about a 2 to 2.5 percent house advantage. Poor players face house advantages of 5 percent or more. They should find another way to spend their money.”
Watson grinned at Holmes’ deadpan expression. Then he noticed a crowd around one table, cheering loudly, slapping each other’s hands and raking in the chips.
“What about that game, Holmes? Will you blend that into your solution?”
Holmes permitted a hint of a smile across his lips.
“Of course, Watson. Are you familiar with craps?”
“I know that it is played with two dice. Other than that, I know every little.”
“A shame, Watson. That is a game I will certainly recommend to the American. There are several wagers that fall well within the 2 percent solution. One is called the pass line. It starts with what is called the ‘comeout roll.’ If the shooter rolls a total of 7 or 11 on the two dice, those who wager on the pass line win, and if he rolls a 2, 3 or 12, they lose. If it is any other number, it becomes the ‘point.’ If the shooter rolls the point number again before rolling a 7, pass bettors win. If the 7 comes first, they lose.”
“That sounds deucedly complex, Holmes.”
“Do not trouble yourself, Watson. For those who play, the details become second nature quickly enough. You should note there is a bet called ‘come,’ which is decided in the same way. The only difference is that it is made when there is already a point number for pass line wagers. The house has a 1.41 percent advantage on either pass or come. Further, there ware wagers called ‘don’t pass’ and ‘don’t come’ that work in nearly the opposite way. They win when pass and come lose, and lose when pass and come win, except that a 12 on the first roll on don’t pass and don’t come just returns money to the player, with no winnings. On these, the house has an advantage of 1.4 percent.”
“I see, my dear Holmes. All well within the 2 percent solution.”
“Indeed, Watson. Such is also the case for what are called ‘place bets’ on the numbers 6 and 8. If the player places the 6, he wins if the shooter rolls a 6 before rolling a 7. If the 7 comes first, the bettor loses. The house advantages on either 6 or 8 are 1.52 percent.
“However, Watson, there are many other possible wagers on the pass line, with much higher advantages to the house. My correspondent would be wise to stick to pass and come, don’t pass and don’t come, placing the 6 and 8, and pretending the other possiblities, tempting as they may be, don’t even exist.”
Watson pulled out his notebook and pen, and checked off a list.
“Baccarat, blackjack, craps. Is there anything else you would like to show me, Holmes?”
“Patience, Watson. We have just a couple of more stops. Do you see the game Three Card Poker?”
“It is really two games in one. One is ante-bet, in which the player’s hand must beat the dealer’s, and the other is Pair Plus, in which all hands of a pair or better are paid. We shall disregard Pair Plus as being just outside our parameters. There are several versions of the pay table, and the best has a house advantage of 2 percent. However, in ante-bet, we can look at the house advantage in two ways. If we regard the house advantage as the percentage of the ante, which players must make to start play, that he house expects to keep, it is 3.4 percent. However, if we regard it as the percentage the house expects to keep of the ante plus the optional bet players make if they like their cards, then the advantage is 2.01 percent.
“Even I shall not draw so fine a line to keep the game outside the 2 percent solution by a margin of one one-hundredth of a percent.”
“Why Holmes,” Watson said with a chuckle. “I should never have expected you to be so cavalier on a matter of precision.”
“Tend to your notepad, doctor,” Holmes said matter of factly. “There is one more stop we must make.”
The direction caught Watson by surprise.
“We’re walking back toward the slot machines, Holmes. I thought you said they were not part of the 2 percent solution.”
“They are not, my dear Watson. And I fear that for my American friend, neither are these.”
He pointed to a large number of video poker games.
“If you are willing to learn the necessary strategies, and to look at the pay tables and sort the best games from the worst, many video poker games fall within that 2 percent threshold. A select few even give an advantage to an expert player. However, our correspondent wrote that he is not willing to put in the necessary time. So alas, we must leave casing out the best of these for the moment.”
“So Holmes, what we have in the 2 percent solution are baccarat, blackjack, craps, Three Card Poker, perhaps video poker for another client. If I may make so bold, what game would you choose? No, don’t tell me. Let me guess. You would be a counter of cards, applying your skills of observation and analysis to gain an advantage.”
Holmes grinned broadly at last.
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