Our title comes from a famous book by Lester Dubbins and Leonard Savage, which appeared at the beginning of the Bayesian Theoretical Resurgence (late 1970s), a movement which has by now infiltrated nearly all of academia. The next logical step (a pun!) on this road to a complete understanding of uncertainty is full-on logical probability. Academia is now far too distracted to venture down that road, so this trip will be some time coming.
Games of chance are always logical. Prove this by picking up any undergraduate text in statistics and find the chapter on probability. You will see examples like this: “A die has probability 1/6 of showing a 6; therefore, the probability of two die (or one die thrown twice) showing two 6s is 1/6 x 1/6 = 1/36.”
Valid answer, but an invalid, or at least incomplete, argument. Close enough, however, because of the implicit recognition that logic allowed us to deduce the probability of the die showing 1/6. Which comes from the statistical syllogism: an object has six states, and only one of which must obtain; therefore, the probability it is in any state is 1/6. Notice we didn’t have to say a “fair” or “unweighted” object! Once that deduction is in hand, we can prove theories of what will happen to that die (or object) under named scenarios, which come in two flavors.
Simple gambles are those where the premises of the game allow us to deduce the probabilities of events of interest. Examples: “Two 6s on two throws”, “A pair of jacks showing from a poker deck”, “00 showing on a roulette wheel”, “Three cherries showing on a slot machine”, and so on. Casinos provide simple gambles.
Complex gambles are different from simple ones because we cannot deduce the probability of the events of interest. Events do not have probabilities! We can only deduce probabilities after premises are given. For these events, we can’t (usually) agree on all the premises of the gambles. For instance, what are the premises for these gambles? “The person to my left in this poker game holds a hand superior to mine”, “Horse A will win the race or at least come in third”, “Stock B will increase in price over the next week”, or “The Detroit Tigers will win tomorrow’s game.”
We don’t know: nobody knows. The ideal premises are those which give us the complete cause of each event. We haven’t any idea how to get the complete causes of such complex events. Best we can hope for are, maybe, some causes, and good correlations—which recognize cause somewhere in a chain of causes of the event.
Now, unless you are the owner of a casino, or are a bookie, it is impossible to consistently (the key word) make money with simple gambles. You might, but probably will not, consistently make money with complex gambles. Unless you’re good at identifying causal or cause-like premises. Here’s why: because all know all the relevant premises for simple gambles, but not for complex gambles.
Any casino game that does not involve the intelligences of other human beings can be analyzed as simple gambles. This means we can, without error, compute the probability of any outcome of any game, which we can call A, for example A = “The roulette wheel shows red” given the premises of the gamble. We will always know, given the properties and setup of the game, the probability A will be true, because we know the premises. For ease, call that probability Pr(A|gamble premises), which I emphasize all know. (Or should know: not every gambler is savvy.)
It costs you D dollars to bet on A, and you will win with probability Pr(A|gamble premises) and will be paid W dollars if A happens.
The casino sets the required bet D so that it is more than W x Pr(A|gamble premises) (alternatively, W is set less than 1/Pr(A|gamble premises) for every dollar bet). They do this on all simple gambles.
An example: if A = “7 shows on an American roulette wheel” then we can deduce that Pr(A|gamble premises) = 1/38 (the numbers 0-36 and the symbol “00” are on the wheel). It costs (say) 1 Dollar to play. If you win, you receive 35 Dollars. In this case, W x Pr(A|gamble premises) = 35/38 which is less than the 1 Dollar it costs to play. On average, the casino takes in 8 cents for every dollar bet, meaning you lose 8 cents on average.
This “on average” is also a deduction, given the gamble premises. We can calculate the exact probabilities of winning for the casino given an assumed number of bets and amounts; the probability the casino is in the black is nearly 1 for even a modest number of bets.
Meaning you will go hungry if you make gambling on these games a career. Exceptions are gambles like blackjack, where strategies exist to edge the bet in your favor. But that is because D < W x Pr(A|gamble premises) is not equivalent to D < W x Pr(A|card sequence & gamble premises). Indeed, it can be that Pr(A|card sequence & gamble premises) > Pr(A|gamble premises), for instance, in card counting schemes. If there is a single deck of cards (which is rare), and you have seen all the aces have already been played, then, if you are paying attention and are good at simple math, Pr(A|card sequence & gamble premises) > Pr(A|gamble premises).
You can make money. But because casinos have more money than you, and politicians desire to have that money, casinos are able to buy laws that make these card-counting strategies illegal. Just as you go to Walmart to purposely part with your cash, you are meant to go to a casino to lose money.
Money can be made with complex gambles. That doesn’t mean it’s easy, only that it can be done. In simple gambles, everybody has the same information about Pr(A|gamble premises). This isn’t true in complex gambles where to win, you need to have better information (premises) about A than the person or persons betting against you. That is, Pr(A|my gamble premises) ≠ Pr(A|your gamble premises), except only rarely when you and I agree on all premises—actual, tacit, and implicit.
That all these three sets of premises exist, and often can scarcely be articulated, is why probability is sometimes thought to be subjective, which it isn’t. Because if we agree on all premises, we must necessarily agree on the probability.
Back to complex gambles. Those cigarette-wielding guys huddled around the Off Track Betting entrance aren’t just dosing themselves with nicotine. They’re trying to gain an advantage in information by subtle probes of their compatriots. Brokerages ponder quarterly reports for the same reason. And so on. It’s all gambling.
Problem is, everybody else is trying to gain an edge the same time you are, which usually means your information is not much better than the next person’s. Plus, in betting horse races and the stock market, there are those darn transaction fees. Tracks skim a percent off the top and set the payouts by the amounts bet, making it extremely difficult to win money. Brokers and banks charge transaction fees which cause the same difficulties.
Besides bar bets, insider trading, and outright cheating, a gamble with real potential is poker (and its business equivalents) which depends on bluffing and the ability to detect it. Being able to read tics and tells of others is a great skill; but it’s a rare talent and expensive to acquire.
The lesson is: stay away from simple gambles and only take bets where you are sure of your information.
This is an update and clarification to a post which originally ran in June 2009, written before I had come to a full, or at least fuller, understanding on probability.
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