7
P
robability is one of the most useful quantitative techniques available to public and nonprofi t managers. Probability helps a manager know how likely it is that certain events will occur. Using the rules of probability as discussed in this and the following chapters, public and nonprofi t managers can diagnose and solve a variety of problems. For example, the head of maintenance in a city could use probability to determine how frequently major breakdowns will occur in the city’s automobile and truck fl eet, and could then schedule main- tenance personnel accordingly. A fi re chief could use probability to allocate her workforce to the areas of the city where most fi res occur. An affi rmative action offi cer could use probability to determine whether an agency discriminates in its hiring practices. A nonprofi t manager could use probability to forecast demand for holiday meals to the homeless in a city. Another nonprofi t agency might use probability to determine which areas of the county are most in need of services for the homebound elderly.Th is chapter covers the basic rules and assumptions of probability. Th ese rules and assumptions will permit us to treat some advanced probability techniques in the next two chapters.
Step 3: Divide the fi rst number by the second; the answer gives you the probabil- ity that the event in question will occur.
Let us look at some examples. Suppose you have an unbiased coin (an unbiased coin is one that has an equal probability of coming up heads or tails). What is the probability that the coin will land head side up if fl ipped? Th e number of possible ways a head can appear is one; there is only one head per coin. The number of total possible outcomes is two; a coin fl ip may come up either a head or a tail. Th erefore, the probability of obtaining a head when fl ipping a coin is 1 divided by 2, or .5. Figure 7.1 shows the probability tree for one fl ip of a coin.
A probability tree diagrams a sequence of events or outcomes with the associated probabilities of their occurrence.
Th e probability of rolling a 6 on one roll of an unbiased die is .167. Th ere are six sides to a die, all of which are equally likely to land that side up. Only one of those sides has a 6 on it, so the probability of rolling a six is 1 divided by 6, or .167. Th e probability of drawing a spade from a deck of 52 playing cards is the number of spades (13) divided by the total number of cards, or .25. Because a deck of playing cards has four suits each equally likely to occur, the probability of .25 (one out of four) makes good sense.
Th e same logic applies to events with more than one trial (that is, a fl ip of a coin, or a roll of a die, or a draw of a card). For example, what is the probability of obtaining two heads if an unbiased coin is fl ipped twice? Th is probability may be determined by a probability tree. On the fi rst fl ip, a coin may land on either a head or a tail, as shown in Figure 7.1. If the result was a head on the fi rst fl ip, the second fl ip could be either a head or a tail. If the coin was a tail on the fi rst fl ip, then on the second fl ip it could be either a head or a tail. Th e probability tree in Figure 7.2 shows all possible results of two fl ips of a coin.
Examining the probability tree in Figure 7.2, you can see that the number of ways that two heads will appear is 1. Th e number of ways that all possible out- comes can occur is 4. Th erefore, the probability of obtaining two heads on two fl ips of an unbiased coin is one out of four, or .25.
The same logic can be applied to more than two trials in any event. For example, in the space provided on page 115 draw the probability tree for all pos- sible outcomes when a coin is fl ipped three times.
Figure 7.1 Probability Tree for One Flip of a Coin
Heads
Tails Flip No. 1
What is the probability that you will obtain three heads on three consecu- tive fl ips of a coin? If your answer was .125 (or 1 4 8), congratulations. If not, recheck your probability tree.
Using this same tree, determine the probability of obtaining two heads on three fl ips of a coin. Examining the tree reveals that the probability of two heads is three out of eight, or .375 (three combinations: head-head-tail, head-tail-head, or tail-head-head).
What is the probability of obtaining exactly two tails on three fl ips of a coin?
Using the probability tree, you should fi nd a probability of .375 (or three out of eight). What is the probability of obtaining three tails on three consecutive fl ips of an unbiased coin? Th e answer is one out of eight, or .125.
What is the probability of obtaining two or more heads on three fl ips of an unbiased coin? In this situation, you would add the number of ways of achieving three heads (one) to the number of ways of achieving two heads (three) and divide this number by the total number of possible outcomes (eight), for a probability of .5 (or four out of eight).
Finally, what is the probability of obtaining zero, one, two, or three heads on three fl ips of an unbiased coin? Because these are all the possible events, the answer is 1.0. Th is is one way to check that your probabilities are correct. Th e probability for all possible events should be 1.0.
Figure 7.2 Probability Tree for Two Flips of a Coin
Outcome
Heads
Tails
Flip No. 1
Heads Heads, Heads
Tails Heads, Tails
Flip No. 2
Heads Tails, Heads
Tails Tails, Tails
Flip No. 2
Let us assemble these probabilities into a probability distribution.
A probability distribution tabulates the probability of occurrence of each event or outcome in a particular domain. For example, the tables presented in the Statistical Tables at the end of this book are probability distributions. Table 7.1 presents a much simpler example, the probability distribution of the number of heads occurring in three fl ips of an unbiased coin. As the table shows, the basic law of probability is used to calculate the probability of each outcome (number of heads). In the table, H represents heads, and T corresponds to tails. You can use the probability distri- bution to check your answers to the questions asked above.
Now let us consider some defi nitions. Any probability that can be deter- mined logically before an event actually occurs is called an a priori probability.
Th e examples just given are a priori probabilities. With a priori probabilities, you do not have to carry out the experiment to determine the probabilities. By contrast, probabilities generated by numerous trials are called posterior prob- abilities. For example, if we do not know whether a coin is unbiased, we can fl ip it numerous times to fi nd out. Th e ratio of heads to total fl ips is the posterior probability of fl ipping the coin and obtaining a head. An unbiased coin has a probability of .5 of coming up heads. Posterior probabilities may also be called long-run probabilities.
Although a priori probabilities present a good introduction to the subject, most of the probabilities that public and nonprofi t managers calculate and use are posterior probabilities. In contrast to a priori probabilities, in which we know logically, for example, that the probability of obtaining a head in an unbiased coin fl ip is .5, with posterior probabilities there is no way to know the probabili- ties in advance. Rather, we must obtain the relevant data and calculate the relative frequency of occurrence of the events of interest. For example, the probability of employees of the city public works department earning a promotion within 2 years of hiring, that of their requesting a transfer to another department, that of their having a statistical package program loaded on their desktop computer at work, and that of their referring to this book to help them interpret results from it—all are posterior probabilities. Th e probability that a nonprofi t agency in the state will have more volunteers than paid employees, that it will receive funding
Table 7.1 Probability Distribution of Number of Heads in Three Flips of a Coin Outcome: Number
of Heads
(1): Number of Ways Outcome Can Occur
(2): Number of
Possible Outcomes (1) 4 (2):
Probability
0 1 (TTT) 8 .125
1 3 (HTT; THT; TTH) 8 .375
2 3 (HHT; HTH; THH) 8 .375
3 1 (HHH) 8 .125
1.000
from government, and that it will have interns from MPA programs are also pos- terior probabilities. Th ese probabilities can be determined only by obtaining the relevant information from agency records, observation, surveys, and so forth and by performing the necessary calculations. Later in this chapter we show how to apply the rules of probability to the calculation and interpretation of posterior probabilities.