What is Probability?

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Chapter: Biostatistics for the Health Sciences: Basic Probability

Probability is a mathematical construction that determines the likelihood of occurrence of events that are subject to chance.


WHAT IS PROBABILITY?

Probability is a mathematical construction that determines the likelihood of occurrence of events that are subject to chance. When we say an event is subject to chance, we mean that the outcome is in doubt and there are at least two possible outcomes.

Probability has its origins in gambling. Games of chance provide good examples of what the possible events are. For example, we may want to know the chance of throwing a sum of 11 with two dice, or the probability that a ball will land on red in a roulette wheel, or the chance that the Yankees will win today’s baseball game, or the chance of drawing a full house in a game of poker.

In the context of health science, we could be interested in the probability that a sick patient who receives a new medical treatment will survive for five or more years. Knowing the probability of these outcomes helps us make decisions, for ex-ample, whether or not the sick patient should undergo the treatment.

We take some probabilities for granted. Most people think that the probability that a pregnant woman will have a boy rather than a girl is 0.50. Possibly, we think this because the world’s population seems to be very close to 50–50. In fact, vital statistics show that the probability of giving birth to a boy is 0.514.

Perhaps this is nature’s way to maintain balance, since girls tend to live longer than boys. So although 51.4% of newborns are boys, the percentage of 50-year-old males may be in fact less than 50% of the set of 50-year-old people. Therefore, when one looks at the average sex distribution over all ages, the ratio actually may be close to 50% even though over 51% of the children starting out in the world are boys.

Another illustration of probability lies in the fact that many events in life are un-certain. We do not know whether it will rain tomorrow or when the next earthquake will hit. Probability is a formal way to measure the chance of these uncertain events. Based on mathematical axioms and theorems, probability also involves a mathematical model to describe the mechanism that produces uncertain or random outcomes.

To each event, our probability model will assign a number between 0 and 1. The value 0 corresponds to events that cannot happen and the value 1 to events that are certain.

A probability value between 0 and 1, e.g., 0.6, assigned to an event has a fre-quency interpretation. When we assign a probability, usually we are dealing with a one-time occurrence. A probability often refers to events that may occur in the fu-ture.

Think of the occurrence of an event as the outcome of an experiment. Assume that we could replicate this experiment as often as we want. Then, if we claim a probability of 0.6 for the event, we mean that after conducting this experiment many times we would observe that the fraction of the times that the event occurred would be close to 60% of the outcomes. Consequently, in approximately 40% of the experiments the event would not occur. These frequency notions of probability are important, as they will come up again when we apply them to statistical inference.

The probability of an event A is determined by first defining the set of all possi-ble elementary events, associating a probability with each elementary event, and then summing the probabilities of all the elementary events that imply the occur-rence of A. The elementary events are distinct and are called mutually exclusive.

The term “mutually exclusive” means that for elementary events A1 and A2, if A1 happens then A2 cannot happen and vice versa. This property is necessary to sum probabilities, as we will see later. Suppose we have event A such that if A1 occurs, A2 cannot occur, or if A2 occurs, A1 cannot occur (i.e., A1 and A2 are mutually exclu-sive elementary events) and both A1 and A2 imply the occurrence of A. The proba-bility of A occurring, denoted P(A), satisfies the equation P(A) = P(A1) + P(A2).

We can make this equation even simpler if all the elementary events have the same chance of occurring. In that case, we say that the events are equally likely. If there are k distinct elementary events and they are equally likely, then each elemen-tary event has a probability of 1/k. Suppose we denote the number of favorable out-comes as m, which is comprised of m elementary events. Suppose also that any event A will occur when any of these m favorable elementary events occur and m < k. The foregoing statement means that there are k equally likely, distinct, elemen-tary events and that m of them are favorable events.

Thus, the probability that A will occur is defined as the sum of the probabilities that any one of the m elementary events associated with A will occur. This probabil-ity is just m/k. Since m represents the distinct ways that A can occur and k represents the total possible outcomes, a common description of probability in this simple model is


P(A) = m/k = {number of favorable outcomes} /{number of possible outcomes}

Example 1: Tossing a Coin Twice. Assume we have a fair coin (one that favors neither heads nor tails) and denote H for heads and T for tails. The assumption of fairness implies that on each trial the probability of heads is P(H) = 1/2 and the probability of tails is P(T) = 1/2. In addition, we assume that the trials are statisti-cally independent—meaning that the outcome of one trial does not depend on the outcome of any other trial. Shortly, we will give a mathematical definition of statis-tical independence, but for now just think of it as indicating that the trials do not in-fluence each other.

Our coin toss experiment has four equally likely elementary outcomes. These out-comes are denoted as ordered pairs, which are {H, H}, {H, T}, {T, H}, and {T, T}. For example, the pair {H, T} denotes a head on the first trial and a tail on the second. Because of the independence assumption, all four elementary events have a proba-bility of 1/4. You will learn how to calculate these probabilities in the next section.

Suppose we want to know the probability of the event A = {one head and one tail}. A occurs if {H, T} or {T, H} occurs. So P(A) = 1/4 + 1/4 = 1/2.

Now, take the event B = {at least one head occurs}. B can occur if any of the el-ementary events {H, H}, {H, T} or {T, H} occurs. So P(B) = 1/4 + 1/4 + 1/4 = 3/4.

Example 2: Role Two Dice one Time. We assume that the two dice are indepen-dent of one another. Sum the two faces; we are interested in the faces that add up to either 7, 11, or 2. Determine the probability of rolling a sum of either 7, 11, or 2.

For each die there are 6 faces numbered with 1 to 6 dots. Each face is assumed to have an equal 1/6 chance of landing up. In this case, there are 36 equally likely ele-mentary outcomes for a pair of dice. These elementary outcomes are denoted by pairs, such as {3, 5}, which denotes a roll of 3 on one die and 5 on the other. The 36 elementary outcomes are

{1, 1}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 1}, {2, 2}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 1}, {3, 2}, {3, 3}, {3, 4}, {3, 5}, {3, 6}, {4, 1}, {4, 2}, {4, 3}, {4, 4}, {4, 5}, {4, 6}, {5, 1}, {5, 2}, {5, 3}, {5, 4}, {5, 5}, {5, 6}, {6, 1}, {6, 2}, {6, 3}, {6, 4}, {6, 5}, and {6, 6}.

Let A denote a sum of 7, B a sum of 11, and C a sum of 2. All we have to do is iden-tify and count all the elementary outcomes that lead to 7, 11, and 2. Dividing each sum by 36 then gives us the answers:

Seven occurs if we have {1, 6}, {2, 5}, {3, 4}, {4, 3}, {5, 2}, or {6, 1}. That is, the probability of 7 is 6/36 = 1/6 0.167. Eleven occurs only if we have {5, 6} or {6, 5}. So the probability of 11 is 2/36 = 1/18 0.056. For 2 (also called snake eyes), we must roll {1, 1}. So a 2 occurs only with probability 1/36 0.028.

The next three sections will provide the formal rules for these probability calculations in general situations.

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