16.1: What do the Numbers Mean?

You can now calculate the probabilities of various things happening. But what do the numbers you get tell you—what do they mean? The answer is that they mean different things in different cases. We will note three important cases, and the answer is different for each of them.

Ratios of Successes to Failures

With common games of chance, we can determine the probabilities of simpler outcomes intuitively; indeed, it is much easier to do this than it is to calculate them with the relevant rule. Let’s analyze what we do when we make these intuitive determinations. You are going to draw one card from a full deck. What is the probability that you’ll draw a king? You didn’t need any complicated rules to answer this. Instead, you reason that: there are four kings out of 52 cards, and we are equally likely to draw any one of them, so the probability of getting a king is 4/52.

In such cases, where each of the outcomes is equally likely to occur, we take the number of outcomes of interest to us, divide it by the total of possible outcomes, and interpret this ratio as a probability.

Number of outcomes of interest = number of outcomes of interest / number of all possible outcomes

For example, in the case of drawing a king from a deck, the outcomes of interest are getting a king, and there are four of these. And the set of all possible outcomes consists of drawing any of the 52 cards in the deck. If we call the outcomes of interest a success (terminology that goes back to the gambling roots of probability), we can say that the probability of a success is:

# Successes / # Possible cases

A similar approach works for outcomes of drawing jellybeans from a jar, throwing dice, spinning a roulette wheel, and the like.

But this approach only works when the basic cases of interest are equally likely. It works when we flip a fair coin; the probability of heads is the numbers of cases of interest, or the number of cases of interest, successes as they are often called, over the number of possible cases. There is one way to flip a head and two possible outcomes. So, the probability is 1/2. But this doesn’t work if we flip a biased coin, say one that is twice as likely to come up heads as tails. There is still just one way to have a success (i.e., to flip a heads) and just two possible outcomes (heads and tails), but the probability of a head will no longer be 1/2. To handle cases like this, and many other real life cases as well, we need to turn to frequencies.

Frequencies

In many cases, probabilities are empirically determined frequencies or proportions. For example, the probability that a teenage male driver will have an accident is the percentage or frequency of teenage male drivers who have accidents. This approach applies, though sometimes less clearly, to the price of insurance premiums, weather forecasting, medical diagnosis, medical treatment, divorce, and many other cases.

For example, a health insurance company records the frequency with which males over 50 have heart attacks. The company then translates this into a probability that a male over 50 will have a heart attack, and charges accordingly for the policy. Again, when your doctor tells you that there is a 5% chance that a back operation will worsen your condition, they are basing their claim on the fact that about 5% of the people who get such operations get worse. The outcomes of interest (getting worse) divided by the total number of cases (all those having this kind of surgery) is 5/100.

In many cases, some of the possible outcomes are more likely to occur than others, but we can adapt the basic approach by viewing the probability of a given sort of event as the relative frequency with which it occurs (or would occur) in the set of possible outcomes.

Degrees of Belief

Often, we do not have access to solid information about frequencies, and sometimes it isn’t even clear which frequencies are relevant. But even in these cases, we often have beliefs that involve something very like probabilities. For example, we don’t have solid information about frequencies that would let me assess the probability that aliens from outer space have infiltrated the college’s golf team. Nevertheless, we believe that probability to be very low. Or, to take a more serious example, if you serve on a jury, you may have to form a judgment about the likelihood that the defendant is guilty.

It may be unclear how we can assign a probability to the statement, “Aliens from outer space have infiltrated the golf team” (let’s abbreviate this as A). But whatever rough probability value we assign it, our beliefs will only cohere with each other if we assign further rough probabilities in accordance with the rules of probability.

For example, since we think that the probability of A is very low, we believe that 1- Pr(A) of its negation is very high. And we believe that Pr(A or ~A) = 1 and that Pr(A & ~A) = 0.

In short, probabilities sometimes represent ratios involving equally likely cases, they sometimes represent frequencies, and they sometimes represent our degrees of belief. The former is much easier to work with, but many things that matter in life involve the second or third. Fortunately for us, these issues don’t matter a lot in the sorts of cases we are likely to encounter.

How Can We Comprehend Such Tiny Numbers?

We can develop some feel for the meaning of frequency probabilities when they aren’t too small. For example, the probability that you will roll a two with a fair die is 1/6. This means that on average, over the long run, you will roll a two one-sixth of the time.

But many probabilities are much smaller numbers. For example, the probability of getting kings on two successive draws from a full deck when we replace the first card is 4/52 x 4/52 (approximately .0059), whereas the probability of getting two kings when we don’t replace the first card is 4/52 x 3/51 (approximately .0045). We aren’t used to thinking about such tiny numbers, and it is difficult to get a grip on what they mean. In a highly technological world, the differences between numbers like this are sometimes important, and they also matter to casinos that want to stay in business. But such differences don’t matter much to us in our daily life, and we won’t agonize over them. The important point for us is that most of us have a poor feel for very large and very small numbers, even in cases where their relative sizes are very different.

We have considered the probabilities of outcomes when we draw cards or roll dice, but people also consider the probabilities of outcomes in cases that matter a lot more, including matters of life and death. What is the likelihood of dying in a plane crash? Of getting cancer if you smoke? Of contracting HIV if you don’t use a condom?

Terrorism is frightening and continued to occupy a relatively large portion of American news and public discourse over a decade after 9/11. In fact, however, far fewer than one in a million Americans are killed by terrorists in any given year, whereas over one in 5,000 are killed in automobile accidents. The differences between the probabilities of these two occurrences is enormous, and any rational assessment of how we live our lives should take this into account.

If we had a good feel for large numbers, we could apply this to probability; for example, it would give us a better feel for the magnitude of the difference between 1/5000 and 1/1,000,000. But most of us are no better with big numbers than with small ones. When we hear about the size of the national debt, which is measured in trillions of dollars, the numbers are so enormous that our minds just go numb. A good way to develop some feel for the meanings of very large and very small numbers is to translate them into concrete terms, ideally into terms that we can visualize. What does one thousand really mean? What about ten thousand? Well, the Straz Center for the Performing Arts seats less than five thousand (4,327), the Amalie Arena seats just over twenty thousand (20,500) and the Raymond James Stadium seats around sixty-five thousand (65,890).

With larger numbers, visualization becomes difficult, but analogies can still be useful. Consider the difference between one million (1,000,000) and one billion (1,000,000,000). It takes eleven and a half days for one million seconds to elapse, whereas it takes thirty-two years for one billion seconds to tick away (how long does it take for one trillion—1,000,000,000,000— seconds to elapse?). And the relative difference in probabilities of one in a million and one in a billion is equally immense.

Exercises

1. If it takes about 32 years for a billion seconds to elapse, how long does it take for a trillion seconds to elapse? Explain how you arrived at your answer.
2. How can we apply the points we have learned about the differences between a million and a billion to the claims that one alternative has a chance of one in a million of occurring and a second alternative has a chance of one in a billion of occurring?
3. Can you think of any concrete image that could help you get an intuitive handle on the number 1,000,000? Give it your best shot

Probabilistic Reasoning without Numbers

In our daily lives, we rarely worry about precise probability values; indeed, such numbers are often unattainable or even meaningless. But in the next few sections, we will see that the concepts we acquired in mastering the rules of probability will help us understand many things that happen in real life. We will see how probabilistic concepts are relevant, even in the absence of precise numerical values for probabilities.