Two Parts of a Bet

A betting proposition can not be evaluated as “good” or not unless the bettor has three items of information, corresponding to the three elements of a bet: (1) the risk (the amount of the bet), (2) the reward (the prize for winning), and (3) the contingency on which winning depends. In Chapter One there was a hint about how one might define a bet as “good” in this context: The Type I player might think any bet is “good” if it is interesting or exciting; the Type II player will need to know more.

Part One of the Bet: How Likely?

For the Type II player, “good bets” are those in which the expectation from the contingent event is higher than the risk he or she must take to obtain the bet.

This is a dense sentence, but it’s basic to all gaming. So let’s parse it.

The element that first attracts our attention is the third one, “expectation from the contingent event.” For purposes of clarity, let’s call this element the “physical expectation,” so as to distinguish it from the financial expectation or expected profit from the bet. Although physical expectation is a precise, mathematical concept and term, it can be thought of in an informal way as what is “expected” to happen in the real world. (Technically, it is the expected value, or mean (or average or high point) of a probability distribution).

Suppose the proposition is that the Browns and the Bears won’t score more than 40 points between them in Sunday’s game. Who knows? Well, some sporting expert related to or hired by the casino has probably done a lot of research and some analytical thinking (we presume) and come up with the idea that 40 is about what to expect in point total out of both teams. (We’ll get to the precise syntax of this sort of bet in the discussion of sports books, later).

Suppose you as a wagerer hardly know that the Browns and the Bears are professional football teams. There is no reason to suspect, then, that you will have more information or bet-ter analysis than the person who put together this proposition in the first place. You have no way of knowing whether 40 points seems low, high or about right. You do not have information on item 3 of the bet, the “Expectation from the Contingent Event.” If you were to make a bet on this proposition, it would be as a Type I player, i.e., a “fan.”

In sports and race betting, the process of determining the expected physical outcome is called “handicapping.” In poker and other casino games, it is usually referred to as “strategy” and sometimes even as “theory,” which is a dubious use of the term. In such contexts, figuring out the physical expectation of the contingent event requires memory rather than research, or sometimes a little lightening fast calculation in the head.

For example, instead of betting on the Browns and the Bears, suppose instead you were sitting at a poker table, playing Texas Hold’em:

You hold, say the 6 and 8 of spades -- two reasonably small cards of the same suit -- in your hand, and two cards of the same suit are on the table, say the four and the king of spades. One more card is to come, and this is the next-to-last betting round. Two other players are “in” but have not been betting super-aggressively. You need to decide whether to put in $8 to call a fourth round bet and raise or fold your cards.

In this example, the third element of the bet – “expectation from the contingent event” or “physical expectation” – is based not on research, but on calculation and analysis of the other players. If the other players were betting, it means they have a pair at least, probably to the king on the board. The raise is troubling, as it means that perhaps the player had a stronger hand and had been sandbagging earlier to attract more money into the pot before pouncing. In any event, your hand as it is now is a beaten hand. Your decision is either to call (by paying $8 to stay in the game), to raise (paying in $12, hoping to chase them off or sweeten the pot), or to fold (flee now!). If the last card is a spade, you will have a flush and probably win (unless Mr. Sandbagger has a full house or something stronger). If the last card is anything other than a spade, you will lose to any pair higher than 8, or three of a kind. (A flush will beat pairs, three of a kind or a straight – this all comes in proper order later). So, what are the odds of having a spade turn up as the last card?

Try to stick with this. It can get dicey. The deck now has, let’s say, 34 cards in it (if the game started with 7 players, that’s 14 cards, plus the 4 on the board, subtracted from 52). You know about 4 spades. That leaves 9 more in the deck. If no one else got a spade as one of the face down (“hole”) cards, your chance of a spade would be 9/34. Assuming ¼ of the cards originally dealt face down (including yours) were spades, that would be 3½ spades (one-fourth of 14). You already have two spades, so assume there are only 7½ spades left in the dealer’s hands. 7½ divided by 34 yields 22% chance that the next card will be a spade.

This example shows that calculating the “expectation” of reality can be complex, even if it is just thinking rather than knowing about the Browns or studying horses. We will need to return to this example again, so keep it in mind.

A third illustration of knowing the “physical expectation” comes from roulette. Most roulette wheels in Las Vegas have 36 little slots with numbers on them, alternating red and black, and one slot named “0” and another slot named “00,” both of which are green. Sup-pose you give yourself a betting proposition. Your lucky number is 27. What are the odds that the little ball will stop exactly on the number 27?

That’s pretty easy. The answer is 2.63%, derived from dividing 1 (number of desired out-comes) by 38 (number of possible outcomes). The odds are one in 38 that the ball will land in that slot.

Now suppose the payoff is 35:1, which would be the wheel odds if there were no “0” or “00.” Is that a “good bet”? This gets a little bit ahead of the narrative, but the point is that if you’re playing roulette to win, it is not a good bet. If you’re playing roulette to see what happens with your lucky number, then the mathematical expectation concerning the out-come really does not matter. You’re back to Type I again.

So to summarize the progress made so far, of the two parts of a bet, the first is to have in-formation about the expectation concerning the contingent event. What might happen, and how likely is it? Sometimes this is pure math. In craps and roulette, for example, players are supposed to keep those probability numbers in their heads. Whether thinking people play these games at all depends considerably on what the payoffs are for the bets.

Sometimes knowing the physical expectation from the contingent event is accomplished by research, as in the case of a football game. Sometimes it is an evaluation of scenarios accompanied by some calculations, as in a Texas Hold’em decision. Whatever it is, if you make a bet without having this piece of information, you are a Type I player. This is not a bad thing. Just drop the pretense or aspiration of being in Type II. You’ll be happier.

Part Two of the Bet: Who has the Advantage?

It’s time to return to the dense sentence we were parsing:

For the Type II player, “good bets” are those in which the expectation from the contingent event is higher than the risk he or she must take to obtain the bet.

So far, we have really talked in non-monetary terms about the contingent event: the chance of getting a 27 in roulette or a spade in poker or more than 40 points in Sunday’s game. Each one of these outcomes is a little artifact of a potential reality that comes at us with a certain probability attached to it. But if you say “yes” to the bet, how does this prediction about the world of three dimensions turn into a money calculation?

In fact, this sentence speaks about “expectation” not as just the physical outcome – three heads, 40 points, a spade, or a 27 – but really as an amount of money attached to that out-come. It is the process of assigning a monetary value to the bet that allows you to know if the bet is “good” in the way a Type II player thinks of the term. This is the second part of the bet. This distinction between “physical expectation” and “financial expectation” is maintained throughout the text.

Another way of looking at the monetizing of the event is to ask whether the bettor has an advantage over the bet taker. Type II players do not make bets when the advantage lies with the house (or adversaries in poker games). So the theme question of this Part Two of the Bet discussion is: Who has the advantage and how big is it?

Most treatises on poker or horse racing or gambling in general focus on this part of the problem, rather than the Part I question, “How likely?” The reason is that the arena for Part One of the bet is the real world. The real world is just too complicated. The big world out there includes all the possible physical outcomes of poker, sports, racing, politics, NASA launches, celebrities and everything else. It does not easily reduce to a book-length treatment of handling probabilities. The authors leave it up to the reader to know or figure out whether the 40 point prop on the Browns and Bears is likely or unlikely. The authors leave it up to the reader to know whether the person to your right in Texas Hold’em really has a pair. (The treatises usually choose not to discuss table games other than blackjack. This concept also comes a little later on).

Treatises do not ignore Part I, of course. There are plenty of books on handicapping and gaming strategy. But they do not give you the answers to your precise question, “what is the physical expectation of this specific, contingent event?” The books handle the Part I issue as a matter of technique, or general concept, trying to improve the reader’s skills at coming up with the answer to the question on his own. They help the reader become a handicapper or a poker strategist. But they can’t answer your specific question. They just try to accelerate your experience, so you’ll be better equipped to find out the answer for yourself.

A book that promises a “system” for winning at any of these games is probably a fraud. “Systems” don’t work because the real world is just more complicated than any “system,” other than the real world itself, of course, which doesn’t make this any easier.

What most books on gaming do concentrate on is how one thoughtfully evaluates whether the bet makes sense, given the bettor’s appraisal of the physical expectation arising from the contingent event.

For example, the books might speak to you thus: “If you think the odds are 46% that the Browns and Bears will not score more than 40 points between them on Sunday, I, the au-thor, will explain to you how to decide if the proposition from the casino is worth betting on.” The books write about this aspect of the subject because many people make unwise bets even when their opinion about the future is solid.

Going back to the original, classic example of the coin toss, if the bettor calculated a 12.5% chance of three heads, and then took the house’s bet, he or she screwed up. In political par-lance, “a mistake was made” because more money was put at risk than the expectation would warrant (zero!).

Part Two of the bet is a matter of wise calculation of the bettor’s advantage, or “edge.” These terms mean the same thing, as well as “win rate,” “profit” or “return on investment.” In simplest terms, if the edge is negative, then the bettor does not have any financial advantage in making a bet. The advantage is with the house (or the adversaries, in Poker). If the edge is positive, then the amount by which it is positive becomes a measure of the desirability of making a bet. Most thoughtful Type II players have a minimum requirement in this department, a subject that is covered in Chapter 4.

Now, the calculation of edge or advantage can be pretty complex some times. Take the ex-ample of the Texas Hold’em poker drama. Remember it? You have two small spades face down, two spades (one big) face up, one card to go, and two others in the game, both of whom are “in” on the last round of betting. It’s going to cost you $8 to stay in the game to see your last card, which has a 22% chance of being a spade. Part One of the bet has been taken care of. That’s the 22% figure. Should you play or fold?

Remember the other two elements of a bet besides the physical expectation? They are the amount at risk and the amount of reward. This is where those elements come into play.

Suppose that the pot now had $52 in it after 3 full betting rounds, with people folding, others raising or otherwise checking and calling. In this round, someone bet and someone raised, which added $12 to make $64 total. (If one of the other players doesn’t fold, there will be $4 more in before the round is over). It’s $8 to you. You already have $12 of your own money in the pot from previous rounds, let’s say. If you fold, it costs you $12. If you play, you’ll have paid $20 for the privilege of being wherever you are. You don’t have to worry about the last betting round, as you will either be in it with a flush or out of it. So the prize will be the $72 or $76 ($52 through round 3 plus $20, perhaps $24 from this round) pot for a price of $8. (Your money bet on previous rounds doesn’t enter into the decision.) This is a payment of 8:1 or better.

Compare a 22% chance of winning with a payout of 8:1. Does it make sense to call the bet to see the last card? Put differently, the chance of winning is 22% and the cost of playing is 12.5% or less. This means that the bettor has a positive monetary expectation, or “edge,” in this bet.

Many poker authorities will remind us that the calculation of “pot odds” might have to include as well what might happen in the next round, if you get your spade. That is to say, you have a 22% chance of making much more than just the pot at the end of this round, but what might happen if more money goes into the pot on the next round? Of course, if you don’t get your spade, you don’t add to the pot, and you have no chance to win it. If we were to take the thinking to that level, the “pot odds” would be even more favorable to the bettor.

The question in this example now becomes, how much of an advantage do I have? To calculate one’s edge (or advantage or profit or win rate or ROI), there is a simple formula.

The first step is to figure out the financial expectation. To do this, multiply each possible outcome in money by the chance of its occurrence and add them together, to cover 100% of all possibilities.

In this case, 22% of $76, say, is $16.72. (The other outcomes have just a zero in them).

The second step is to divide the value of the financial expectation by the price of the bet and subtract 1. The result, expressed as a percentage, is the edge.

In this case, you are paying $8 with an expectation of $16.52. Divide the outcome by the price to play and subtract 1. This yields your percentage edge, which in this case is enormous, over 100% (106.5%)!

One other detail needs to be pointed out from this scenario. By calculating your edge as you have, you are implicitly assuming that the first player, who opened the round 4 of betting, will just call the bets and not raise. If you have stepped into a raising spree between two big hand bettors, then the cost to stay in the game goes up, and the odds that a flush will be beaten anyway also become palpable. In such circumstances, it would be best to fold. So this analysis presumes that you are confident the first player does not have that strong a hand. In fact, he may fold after you call, which leaves the pot at $72. Another question might be, with such a strong edge, why not raise? Many players would do so, as it may cause one or both to fold, it can only increase the pot odds, and even though the chance of winning is less than one in four, the reward for winning is much greater than one in four.

To test whether you’ve got this basic concept solidly in place, first go back to the coin toss. It makes intuitive sense that a 5:1 payout for three heads is a bad deal. Can you articulate why? The edge is negative. To calculate the edge, figure out the finances associated with winning and losing. There is an 87.5% chance of winning nothing, a 12.5% chance of win-ning $5, totaling 100% (It’s always a good idea to check to see if all outcomes are ac-counted for.) The expectation is 62.5 cents. The cost is $1. Divide the outcome by the price to play and subtract 1. That yields -37.5%. The minus sign alone would disqualify the bet, but the 37.5% makes it also a very large negative edge.

Now go to the Browns playing the Bears in Chicago on Sunday. Let’s suppose now that the “prop” from the casino can be restated as follows: $110 bet now will yield $210 later (the $110 of the bet and $100 of winnings) if (according to the side you take) the total of the Browns game does (or does not) exceed 40 points. (The real proposition would be stated a little differently in real life, as described in a later chapter, but this expresses standard terms for a sports bet.)

The house put the point total at 40 in the prop, perhaps where it thinks the odds are about equal for over and under. Then, the house collects $10 net from the losers for going to the trouble of setting up and running the bet. To see this, take the $110 bet of the loser, pay out $100 of it to a winner, (assuming balanced action on the proposition), leaving $10 for the house. This withdrawal of $10 from the betting pool is almost universally called the “vig,” short for “vigorish.” Basically think of the $10 as the fee collected by the house. The vig is a complicating factor, however, and it means that the careful Part 2 bettor will have to sharpen a pencil.

Recall that the “objective” appraisal of our informed bettor was that the chance was 46% that the total points in the game would not exceed 40. The price to play is $110 and the pay-back is $210 (the bet plus the prize). Paying $110 to get $210 yields a 52.38% margin. Any physical expectation greater than this number would make the edge positive. In this case the cost of the bet (52.4%) is greater than the physical expectation from the outcome (46%). So it’s a bad deal. To calculate how bad a deal it is, just for practice, take the percent chance of winning times the amount of the prize (that is, summing the various possible fi-nancial expectations). That yields $96.60. Divide this by the price of the bet and subtract 1. That comes to a minus 12.8% edge or win rate or ROI.

What happens if the score lands right on 40? In most cases, the bettors neither win nor lose. They get their money back. This is called a “push.” (Sometimes the terms of the bet will put the total at a fraction, like 40.5, which effectively rules out this possibility). Let’s sup-pose that the odds of a tie are confidently calculated at 1%. The calculation of edge would then be .46 * $210 + .01 * 110 + .53 * 0 or $97.7. This improves the situation to a negative 11.18% edge, but it’s really no help.

Now, it might be possible to bet on the opposite outcome. According to the above analysis, 53% of the time the total exceeds 40. That wager is available at -110 as well. The edge is .53*210+.01*110+.46*0 or $112.4. This, divided by the price of the wager, $110, and subtracting 1, yields a positive edge of 2.18%. This is not much, but at least it’s in the plus column.

Think about the implications of this wager. To win, one has to have an edge that is at least enough to overcome the inherent “drain” on the bet placed by the house’s vig. The house tries to handicap the wager such that an even amount of action is attracted to each side. If your handicapping yields a confident estimate of the physical outcome that also matches where the house thinks the bet should be, then the proposition will probably be negative or so slimly positive as not to be worth the trouble.

“Get” this next statement, as it is essential to most gaming when the “house” is on the other side of the bet: “It is only when the skilled bettor finds a betting proposition that is enough out of synch with the bettor’s calculation of physical expectation, that edges appear.” If the house is correct on stating the terms of a bet or clever in calculating the payout of a game (such as craps or a slot machine), there will not be any fruitful bets for the Type II player.

This concept can also be applied to poker in the same way. The difference, other than the fact that the adversaries are other gamers, is that the decision to bet does not come in isola-tion, the way it does in a sports or race book or with a slot machine, but rather inlaid into the context of a hand. The rule is the same. Negative expectation bets should be avoided, and positive expectations bets should be played maximally (if the promise is enough positive to make it worth the trouble). As can be seen from the examples given in this chapter, it is common to find very large edges (both negative and positive) in a poker hand, whereas the edges that appear in the sports and race books tend to be smaller, given the intelligent contrivance of the house in setting the lines.

Finally, to come back where we began, as if in a wheel, if you will, let’s consider the roulette proposition. Hitting your number pays 35:1 and the odds are 1 out of 38. Is there any doubt about this bet? The monetary offer of the bet is 2.77% and the chance of success is less, at 2.63%. This is a negative expectation of 5.3%. Roulette is a game for Type I, recreational bettors (and their significant others, sometimes). The math is just not that hard. For practice, let’s lay it out specifically. For a $2 chip you have a reward, if you win, of $72 ($70 of profit plus your bet back.) The physical expectation is 2.63%. Thus, playing for a while would yield an expected income of 2.63% of $72, which is $1.89 per play, on aver-age. The price to play is $2. Dividing the “spoils” by the cost of playing and subtracting 1 yields -5.5%. This is the negative edge inherent in roulette when the wheel has a “0” and a “00” and pays 35:1. For fun, calculate the edge if the wheel only has one green slot. Some of the older wheels (in downtown Las Vegas or elsewhere – not on the Strip) are of this kind.

Summary

It is time to take stock. We have come a long way in relatively few pages. First, recall that there are three parts of a bet, no matter what the bet is about. They are (1) the amount at risk or the price to play; (2) the amount of the reward or pot or jackpot; and (3) the expectation of the actual outcome of the contingent event. Without these three elements, a wager can not be evaluated.

Recall also that gaming is risk-taking. The player has the choice of engaging the brain to win more money than is lost in the effort (Type II player) or of connecting with the emotions and relishing the thrill of just being there (Type I). Most of us are a mixture of the two types most of the time. With one exception, it is only necessary to evaluate the bet if you are a Type II gamer.

Third, remember that there are two parts to every bet. The first is calculating the physical expectation from the contingent event. This is like handicapping the horses, or figuring odds that a card will turn up in poker. The second is comparing that first evaluation with the odds implicit in the bet proposal, that is, the comparison of the price to play with the amount of the prize. If the financial arrangement implicit in the proposition is more favorable than the physical expectation for the outcome, then the bet is “good” in the parlance of a Type II player. The bettor is said to have an edge.

Whether it makes sense to place a bet, and how much of a bet to place, is the subject of Chapter 4. It is not giving too much away to repeat that if there’s no edge, or if the edge is insufficient to overcome the vig or the built-in house advantage or the time value of money, it would still be imprudent to take the risk. Before getting to this important subject, a small detour is necessary into the underlying theory of games of chance. This is the subject of Chapter 3.