What Are the Odds?
Elementary Probability

Sandra LaFave
West Valley College

What does it mean to say there’s a certain probability that an event will occur, or that “the odds” favor or oppose a certain event?

It depends whether you mean

·         Classical Probability, or

·         Relative Frequency, or

·         Subjective Probability

I.  Classical Probability

Also called the a priori theory of probability, it is associated with the mathematicians Pascal and Fermat.

Paradigm: card games in which the deck contains a fixed number of cards

The probability (P) of an event (a) equals the number of possible favorable outcomes (f) divided by the total number of possible outcomes (n).  Or,

The classical theory assumes that all possible outcomes are equally likely, and that we know n.  For example, we know the number of cards in a deck of regular cards (n = 52, for the first draw), and we assume that the deck is fair, i.e., that every card has an equal chance of being drawn. What is the probability that I’ll draw an ace, if there are four aces in the deck?

P(a) = 4/52, or 1/13, or .0769 or 7.69%

Once I’ve drawn a card, and it’s, say, not an ace, the probability that the next draw yields an ace goes up, since n has gone down.

P(a) = 4/51, or .0784 or 7.84%

As n gets smaller and no aces are drawn, the probability of an ace on the next draw increases dramatically.

If all the possible outcomes are known and equally likely, then classical probability yields a definite number every time. It doesn’t generalize; its answer is quite specific. Classical probability doesn’t concern itself with sample size or representativeness.

The results of classical probability obey the Law of Large Numbers. That is, the results will be closer and closer to the percentages predicted the more often we carry out the draw. If we draw the first card from a fair deck one million times, we’ll find that we will get an ace in very close to 7.7% of the cases. This is why casinos consistently make a profit on the games they offer; they know the odds.

II. The Gambler’s Fallacy

Suppose you’re playing a game in which all outcomes are equally likely (e.g., rolling dice), and you are on a losing streak.  You commit the Gambler’s Fallacy if you believe your losing streak makes it more likely that you’ll roll the numbers you want on the next roll (because you’re “due”). The truth is that your odds don’t change; you start over with each roll.

III.  Relative Frequency

We need to use relative frequency probability when the number of possible outcomes (n) is so large we can’t observe all of them. For example, what is the probability that a 60-year-old woman will die of a heart attack within ten years? We can’t observe all 60-year-old women.

But we can predict the probability for random typical women who resemble one another, if we observe enough of them. Here is where we need to watch for the fallacy of hasty generalization, because size and representativeness of the sample really matter here. We need to make sure we observe enough instances (n should be large enough), and the ones we observe need to be typical.

P(a) = observed f / observed n

The probability of 60-year-old women dying of a heart attack within ten years (a) equals the number of 60- year-olds who do die of a heart attack within ten years (f), divided by all comparable 60-year-old women (n).  So if we observed 1000 60-year-old women over ten years, and observed that 35 of them died of heart attacks, we could say the probability of a random 60-year-old woman dying of a heart attack is around 3.5% (35/1000).

P(a) = 35 / 1000

There’s always the chance that our numbers are unreliable because our sample is unrepresentative. But these results are very reliable if the sample is representative and over 1500 (2% margin of error).

Naturally, we can’t predict with certainty for any particular woman. If we could, we wouldn’t need probability theory.

IV.  Subjective Probability

This has more to do with sociology and salesmanship than with statistics. What odds will you take? Those are the odds you get on the subjectivist theory. It’s called subjectivist because it relies only on what people are willing to bet. What odds are offered? Whatever the parties agree to; this isn’t an objective matter. The odds you get when you bet at the racetrack are of this type. The people who run the track make odds they think the customers will find attractive enough to bet on. People who follow racing closely think they are knowledgeable about the horses, but any horse in the race can win. (If the outcome were based on objective probabilities, then a smart player could win consistently.) But the track and the OTB want to make money, don’t they? If you’re willing to bet on a long shot, you can take your chances.

So what kind of probability applies to winning the lottery? To whether the Giants will win the next World Series?