The lot is cast into the lap; but the whole disposing thereof is of the Lord.
How do we interpret probability?
This article is the second in a series on Faith, Probability and Belief, posted in tandem with Bob Drury’s articles on (pretty much) the same topic. For the first, which gives a brief lesson on probability and its application to belief—Pascal’s Wager—see here. In this Part II I’ll discuss several ways of interpreting probability. Let’s look at three of the six interpretations of probability listed in the Stanford Encyclopedia of Philosophy article on interpretations of probability: frequentist, deductive (logical), subjective. The first two were discussed briefly (without being named) in the first article of this series.
Interpreting probability is much like interpreting quantum mechanics—what one thinks quantum mechanics is all about: each interpretation involves the same mathematical framework and each must be consistent with empirical tests of theory. Accordingly, which probability interpretation one takes to be “true” is a matter of personal choice. My choice for interpreting probability is the subjective, for reasons I’ll outline below.
The Deductive (Logical) Interpretation
The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible, or all of which have the same chance.
—Simeon-Denis Poisson (1781-1840), Mathematician and early worker in probability theory, as quoted here.
Poisson’s definition of probability in the quote above is as good a statement of the deductive interpretation of probability as I’ve seen. The notion that all cases are “equally possible” is an assumption of randomness. We illustrated this in the Part I by considering a thrown die, and will further illustrate it in Part III by a discussion of conditional and joint probabilities.
Here’s one other example. What’s the probability of drawing a red face card (red jack, red queen or red king) from a full deck of cards? Here’s my answer: there are 52 cards in the deck. There are four suits: two black suits (clubs and spades), two red suits (diamonds and hearts). There are three kinds of face cards: jacks, queens, kings. Accordingly there are 3×2 red face cards, so the probability of drawing a red face card is simply 6/52 = 3/26.
One problem in using a deductive approach to probability is that the assumption of randomness, equal likelihood for each possible case, may not be correct. If you’re betting on a dealer drawing cards in a random fashion, there may be a card up his/her sleeve or he/she may do a sleight of hand and pull a marked card from the deck. If you assume that each of face of a die is equally likely to show, you may be wrong if the dice are loaded (see Part III). One major problem in using the deductive approach is that one may not be fully knowledgeable about the possible cases, and consequently one needs to proceed empirically, that is to say, one should use a frequentist interpretation.
A Frequentist (Empirical) Interpretation
For frequentists, probabilities are fundamentally related to frequencies of events.
—Jake VanderPlas, “Frequentism and Bayesianism: A Practical Introduction”
The definition given in the above quote can be illustrated by the following example (also see “Conditional Probability and Evidence,” Part III) Suppose you’re at the craps table at a Casino (or in the alley playing an informal game). After 100 throws of the dice, “boxcars” (two dice showing 12) comes up 13 times, and snake eyes (two dice showing 2) not at all. What would you take as the probability of 12 showing up? You’d take the number of events with 12 total and divide by the total number of trials, or 13/100. As shown in the section Conditional Probability and Evidence, the probability that boxcars or snake eyes would turn up, based on a deductive approach and the assumption that the dice were not loaded, would be 1/36, or about 3 times in a 100. The empirical probability of 13 in 100 for boxcars would cause one to suspect that the dice were loaded.
Here’s a problem in using the frequentist approach: one may not be able to take a sufficiently large number of trials to guarantee that the trial sample represents the real situation. Another difficulty, particularly troublesome in polling (which depends on a frequentist interpretation), is bias in the measurement sample: for example, a representative sample of Democrats and Republicans isn’t taken for the poll. And still another problem might arise due to fluctuations, a departure due to chance from the nominal probability (a basketball player with a 90% average for free throws may have a bad day and get only 1 out of 5 attempts.)
Subjective Probability—”The Real Thing”
The probability of an event is the reason we have to believe that it has taken place, or that it will take place.—Simeon-Denis Poisson, as quoted here
The term, “The Real Thing,” in the heading above is the subtitle of Professor Richard Jeffrey’s book, “Subjective Probability.” Most of what I have to say about subjective probability will be drawn from that book, and since it’s available for free on the web, I strongly recommend that the reader go through it. It’s an easy read, and much information about probability and decision making is packed into relatively few pages.
To put it briefly, subjective probability measures belief. It’s quantified by how much one is willing to bet on the proposition. For example (to quote from Jeffrey’s book), if you say the probability that it will rain tomorrow is 70%, that means you’re willing to bet $7 to win $3 (odds: 7/3). Now people don’t always act rationally on their beliefs. They bet in lotteries where the odds are against them, gamble at casinos where the house is always the winner, and buy insurance that makes them a loser in the long run. As the 2017 Nobel Prize Winner for Economics, Richard Thaler, would have it:
Thaler and his behavioralist colleagues, though, correctly note that people are often far from rational, in ways that are essential to understanding human society.
–Robert Graboyes, “The Hill” 10/11/2017
The historical origins of probability, gamblers trying to find the best options for betting, certainly fit the notion of probability as a measure of belief. Moreover, Bayesian methods enable one to modify one’s beliefs, that is to say, introduce new evidence for evaluating probability (see Part III).
Conditional Probability and Evidence
Probability is founded on evidence, or belief in likely evidence. If we say that any face of die is equally likely to be on top after the die is thrown, we believe that the die is fair, symmetrical, not weighted. And, as shown in the first of these articles, the probability of each face coming up will then be 1/6. This probability as a condition of evidence/belief can be expressed symbolically as P(event|evidence/belief). (Expressed as a logical proposition this would be “if evidence then event;” the vertical bar symbol, “| ,” represents an “if … then … ” statement.) So P(#dots|fair die) = 1/6 expresses the fact that each face of the die is equally likely to come up if the die is a symmetrical, unweighted cube.
Now let’s imagine the die isn’t fair, but weighted. Suppose a small extra weight is put behind the one dot face. This would mean the one dot face would be more likely to be on the bottom after a throw. The one dot face is opposite the six dots face, so that would mean the six dots face is more likely to come up than the others. We can say then that P(six dot|weighted one dot) > 1/6. If we weight both dice behind the one dot face, then the probability of getting “snake eyes” will be less than the 1/36 value for unweighted dice and the probability of getting “boxcars” (a total of 12), would be greater.
Accordingly, if you substitute a pair of weighted dice for unweighted, you may overcome the house odds (the usual house payoff would be for a 1/30 probability, i.e. bet $1, get $30 dollars back, including your dollar bet). (Note: you should worry about getting kneecapped if your switch to weighted dice was discovered.)
To sum up (I apologize for being abstract) we can write the conditional probability of an event E, given belief B is true, as P(E | B), which is to say, “if B then E” has a probability P(E | B).
More to Come
In Part III, I’ll discuss Bayes’ Theorem, its relation to subjective probability, and an application to the possible significance of the Anthropic Principle, as it might apply to belief in a creating God.