**Probability Key Definitions—Event Versus Outcome**

Here are definitions of some common terms that will help us understand what we are talking about when we refer to probability.

**Event Versus Outcome**

The terms *event* and *outcome* are easily confused. An event is a single occurrence or trial in the course of an experiment. An outcome is the result of an event.

If you toss a coin 100 times, there are 100 separate events. Each event is a single toss of the coin. If you throw a pair of dice simultaneously 50 times, each act of throwing the pair is an event, so there are 50 events.

Suppose, in the process of tossing coins, you assign ''heads'' a value of 1 and ''tails'' a value of 0. Then when you toss a coin and it comes up ''heads,'' you can say that the outcome of that event is 1. If you throw a pair of dice and get a sum total of 7, then the outcome of that event is 7.

The outcome of an event depends on the nature of the hardware and processes involved in the experiment. The use of a pair of ''weighted'' dice produces different outcomes, for an identical set of events, than a pair of ''unweighted'' dice. The outcome of an event also depends on how the event is defined. There is a difference between saying that the sum is 7 in a toss of two dice, as compared with saying that one of the dice comes up 2 while the other one comes up 5.

**Sample Space**

A *sample space* is the set of all possible outcomes in the course of an experiment. Even if the number of events is small, a sample space can be large.

If you toss a coin four times, there are 16 possible outcomes. These are listed in Table 3-1, where ''heads'' = 1 and ''tails'' = 0. (If the coin happens to land on its edge, you disregard that result and toss it again.)

If a pair of dice, one red and one blue, is tossed once, there are 36 possible outcomes in the sample space, as shown in Table 3-2. The outcomes are denoted as ordered pairs, with the face-number of the red die listed first and the face-number of the blue die listed second.

**Mathematical Probability**

Let *x* be a discrete random variable that can attain *n* possible values, all equally likely. Suppose an outcome *H* results from exactly *m* different values of *x*, where *m* ≤ *n*. Then the *mathematical probability* *p*_{math}(*H*) that outcome *H* will result from any given value of *x* is given by the following formula:

Expressed as a percentage, the probability *p*%(*H*) is:

If we toss an ''unweighted'' die once, each of the six faces is as likely to turn up as each of the others. That is, we are as likely to see 1 as we are to see 2, 3, 4, 5, or 6. In this case, there are six possible values, so *n* = 6. The mathematical probability of any particular face turning up (*m* = 1) is equal to *p*_{math}(*H*) = 1/6. To calculate the mathematical probability of either of any two different faces turning up (say 3 or 5), we set *m* = 2; therefore *p*_{math}(*H*) = 2/6 =1/3. If we want to know the mathematical probability that any one of the six faces will turn up, we set *m* = 6, so the formula gives us *p*_{math}(*H*) = 6/6 = 1. The respective percentages *p*_{math%}(*H*) in these cases are 16.67% (approximately), 33.33% (approximately), and 100% (exactly).

Mathematical probabilities can only exist within the range 0 to 1 (or 0% to 100%) inclusive. The following formulas describe this constraint:

We can never have a mathematical probability of 2, or –45%, or –6, or 556%. When you give this some thought, it is obvious. There is no way for something to happen less often than never. It's also impossible for something to happen more often than all the time.

**Empirical Probability**

In order to determine the likelihood that an event will have a certain outcome in real life, we must rely on the results of prior experiments. The probability of a particular outcome taking place, based on experience or observation, is called *empirical probability*.

Suppose we are told that a die is ''unweighted.'' How does the person who tells us this know that it is true? If we want to use this die in some application, such as when we need an object that can help us to generate a string of random numbers from the set {1, 2, 3, 4, 5, 6}, we can't take on faith the notion that the die is ''unweighted.'' We have to check it out. We can analyze the die in a lab and figure out where its center of gravity is; we measure how deep the indentations are where the dots on its faces are inked. We can scan the die electronically, X-ray it, and submerge it in (or float it on) water. But to be absolutely certain that the die is ''unweighted,'' we must toss it many thousands of times, and be sure that each face turns up, on the average, 1/6 of the time. We must conduct an experiment – gather *empirical evidence* – that supports the contention that the die is ''unweighted.'' Empirical probability is based on determinations of relative frequency, which was discussed in the last chapter.

As with mathematical probability, there are limits to the range an empirical probability figure can attain. If *H* is an outcome for a particular single event, and the empirical probability of *H* taking place as a result of that event is denoted *p*_{emp}(*H*), then:

**Real-World Empiricism**

Empirical probability is used by scientists to make predictions. It is not good for looking at aspects of the past or present. If you try to calculate the empirical probability of the existence of extraterrestrial life in our galaxy, you can play around with formulas based on expert opinions, but once you state a numeric figure, you commit the PF. If you say the empirical probability that a hurricane of category 3 or stronger struck the U.S. mainland in 1992 equals *x*% (where *x* < 100) because at least one hurricane of that intensity hit the U.S. mainland in *x* of the years in the 20th century, historians will tell you that is rubbish, as will anyone who was in Homestead, Florida on August 24, 1992.

Imperfection is inevitable in the real world. We can't observe an infinite number of people and take into account every possible factor in a drug test. We cannot toss a die an infinite number of times. The best we can hope for is an empirical probability figure that gets closer and closer to the ''absolute truth'' as we conduct a better and better experiment. Nothing we can conclude about the future is a ''totally sure bet.''

**Probability Key Definitions Practice Problems**

**Practice 1**

Suppose a new cholesterol-lowering drug comes on the market. If the drug is to be approved by the government for public use, it must be shown effective, and it must also be shown not to have too many serious side effects. So it is tested. During the course of testing, 10,000 people, all of whom have been diagnosed with high cholesterol, are given this drug. Imagine that 7289 of the people experience a significant drop in cholesterol. Also suppose that 307 of these people experience adverse side effects. If you have high cholesterol and go on this drug, what is the empirical probability *p*_{emp}(*B*) that you will derive benefit? What is the empirical probability *p*_{emp}(*A*) that you will experience adverse side effects?

**Solution 1**

Some readers will say that this question cannot be satisfactorily answered because the experiment is not good enough. Is 10,000 test subjects a large enough number? What physiological factors affect the way the drug works? How about blood type, for example? Ethnicity? Gender? Blood pressure? Diet? What constitutes ''high cholesterol''? What constitutes a ''significant drop'' in cholesterol level? What is an ''adverse side effect''? What is the standard drug dose? How long must the drug be taken in order to know if it works? For convenience, we ignore all of these factors here, even though, in a true scientific experiment, it would be an excellent idea to take them all into consideration.

Based on the above experimental data, shallow as it is, the relative frequency of effectiveness is 7289/10,000 = 0.7289 = 72.89%. The relative frequency of ill effects is 307/10,000 = 0.0307 = 3.07%. We can round these off to 73% and 3%. These are the empirical probabilities that you will derive benefit, or experience adverse effects, if you take this drug in the hope of lowering your high cholesterol. Of course, once you actually use the drug, these probabilities will lose all their meaning for you. You will eventually say ''The drug worked for me'' or ''The drug did not work for me.'' You will say, ''I had bad side effects'' or ''I did not have bad side effects.''

Practice problems for these concepts can be found at:

Basics of Probability Practice Test