PROBABILITY Page 1 of 9 I. Probability 1. So far we have been concerned about describing characteristics of a distribution. That is, frequency distribution, percentile ranking, measures of central tendency, measures of dispersion, and relationships between distributions (correlations). 2. However, the interests of scientists go beyond the mere description of data. The fundamental strategy of science is the formulation of general statements about populations or the effects of experimental conditions on criterion variables. 3. That is, I was not merely satisfied to report that the behaviors of ablated siamese fighting fish was different than the behaviors of nonablated siamese fighting fish. More importantly, I wanted to demonstrate that there was some relationship between the frontal lobes and aggressive behaviors and that these results were not just purely chance results. 4. The problem of chance variation is an important one. We all know that the variability of our data in the behavioral sciences engenders the risk of drawing an incorrect conclusion. 5. Let's take a look at the following example: An experimenter hypothesizes that users of marijuana make more driving errors than none users. She administers a driving test to two groups of subjects, four subjects in each group; one receiving the active ingredient of marijuana and the other a placebo. She finds the mean number of errors for the experimental group to be 7 and the mean numbers of errors for the control group to be 4.
PROBABILITY Page 2 of 9 6. Is the experimenter justified in concluding that her hypothesis has been confirmed? The answer is obviously no. But why? After all, there is a difference between the sample means. We might argue that the variability of driving skills is so great and the sample size (N) so small, that some differences in the means is inevitable as a result of our selection procedures. 7. The critical questions which must be answered by inferential statistics then become: Is the apparent difference in the two groups reliable? That is, will it appear regularly in repetitions of the study? Or is the difference the result of unsystematic factors which will vary from study to study, therefore produce sets of differences without consistency? 8. A prime function of inferential statistics is to provide rigorous and logically sound procedures for answering these questions. As we shall see probability theory provides the logical basis for deciding among all the various alternative interpretations of research data. 9. However, before discussing the elements of probability theory, it is desirable to understand one of the most important concepts in inferential statistics, that of randomness. II. The Concept of Randomness 1. A series of events are said to be random if one event has no predictable effect on the next. We can most readily grasp randomness in terms of games of chance,
PROBABILITY Page 3 of 9 assuming they are played honestly. 2. Knowledge of the results of one toss of a coin, one throw of a die, one outcome of the roulette wheel, or one selection of a card from a well-shuffled deck (assuming replacement of the card after each selection) will not aid us one iota in our predictions of future outcomes. This characteristic of random events is known as independence. 3. Only if independence is achieved, can events be said to be truly random. The second important characteristic of randomness is that when the sample is taken from a population, each member of the population should have an equal likely chance to be selected. 4. Thus, if our selections favor certain events or certain collections of events, we cannot justifiably claim randomness. Such sampling procedures are referred to as being biased. 5. If we are interested in American public opinion about the right to an abortion, would selecting people from voters registration be a random sample of American opinion? Or from telephone listing (unlisted numbers, individual who don't own a telephone)? The danger of generalizing to the general population from such biased samples should be obvious to us. 6. It is beyond the scope of this course to delve deeply into sampling procedures since this topic is a full course by itself. Suffice it to say that unless the condition of
PROBABILITY Page 4 of 9 randomness is met, we may never know what population we should generalize our results. Furthermore, with nonrandom samples, we find that many of the rules of probability do not hold. 7. What is nice about randomness is that it appears to bring order to random events. That is, each event may not be predictable when taken alone, but collections of random events can take on predictable form. The binomial distribution illustrates this fact. 8. If we were to take, say, 20 unbiased coins and toss them into the air, we could not predict accurately the proportion that would land heads. However, if we were to toss these 20 coins a large number of trials, record the number turning up heads on each trial, and construct a frequency distribution of outcomes in which the horizontal axis (the abscissa) varies between no heads to all heads, the plot would take on a characteristic and predictable form known as the binomial or Bernoulli distribution. 9. By employing the Bernoulli model, we would be able to predict with considerable accuracy, over a large number of trials, the percentage of the time various outcomes will occur. The same is true with respect to the normal curve model. III. Theories of Probability 1. Probability may be regarded as a theory that is concerned with the possible outcome of experiments. The experiments must be potentially repetitive; i.e., we
PROBABILITY Page 5 of 9 must be able to repeat them under similar conditions. 2. It must be possible to enumerate every outcome that can occur, and we must be able to state the expected relative frequencies of these outcomes. 3. The theory of probability has always been closely associated with games of chance. For example, suppose that we want to know the probability that a coin will turn up heads. 4. Since there are only two possible outcomes (heads or tails) this would be mutually exclusive and exhaustive. That is, it can only be heads or tails, it can't be both. Exhaustive is there is no other possible outcome. That is the coin can't stand on it side. If the coin is unbiased then the probability that heads, p(h), will occur is 1/2. 5. This kind of reasoning has lead to following classical definition of probability: A. Where A=number of outcomes favoring A B. =number of outcomes not favoring A 6. It should be noted that probabilities is defined as a proportion (p). The most important point in the classical definition of probability is the assumption of an ideal situation in which the structure of the population is known; i.e., the total
PROBABILITY Page 6 of 9 number of possible outcomes (N) is known. 7. Although it is usually easy to assign expected relative frequencies to the possible outcomes of games of chance, we cannot do this for most real-life experiments. 8. In actual situation, expected relative frequencies are assigned on the basis of empirical findings. Thus we may not know the exact proportion of students in a university with I.Q.s over 120, but we may study a random sample of students and estimate the proportion who will have I.Q.s above 120. IV. Formal Properties of Probabilities 1. From the classical definition of probability, p is always a number between 0 and 1 inclusively. If an event is certain to occur, its probability is 1; if it is certain not to occur, its probability is 0. 2. In addition to expressing probability as a proportion, several other ways are often employed. It is sometimes convenient to express probability as a percentage or as the number of chances in 100. 3. To illustrate: If the probability of an event is 0.05, we expect this event to occur 5% of the time, or the chances that this event will occur are 5 in 100. V. Probability and Continuous Variables 1. Probabilities require discrete events and doesn't work well with continuous
PROBABILITY Page 7 of 9 variables. But probabilities can work with continuous variables if we convert the continuous variable to area under a curve. 2. Thus, for continuous variables, we may express probability as the following proportion: 3. Since the total area in a probability distribution is equal to 1.00, we define p as the proportion of total area under portions of a curve. VI. Probability and the Normal Curve Model 1. If you remember we said that the standard normal distribution has ì of 0, a óof 1, and a total area that is equal to 1.00. We saw that when scores on a normally distributed variable are transformed into z-scores, we are, in effect, expressing these scores in units of the standard normal curve. 2. This permits us to express the difference between any two scores as proportions of total area under the curve. Thus we may establish probability values in terms of these proportions as in the above formula. 3. What is the probability of selecting at random, from the general population, a person with an I.Q. score of at least 132, assume µ = 100 and ó = 16. The answer
PROBABILITY Page 8 of 9 to this question is given by the proportion of area under the curve above a score of 132. 4. First, we must find the z-score corresponding to X = 132. z= (132-100)/16 = 2.00. In column C (Table A), we find that 0.0228 of the area lies at or beyond a z of 2.00. VII. One- and Two-tailed p-values 1. We posed the question: What is the probability of selecting a person with a score as high as 132? We answered the question by examining only one tail of the distribution, namely, scores as high as or higher than 132. For this reason, we refer to the probability value that we obtained as being a one-tailed p-value. 2. In statistics and research, the following question is more commonly asked: "What is the probability of obtaining a score (or statistic) this deviant from the mean?" 3. Clearly, when the frequency distribution of scores is symmetrical, a score of 68 or lower is every bit as deviant from a mean of 100 as a score of 132. That is, both are two standard deviation units away from the mean. 4. When we express the probability value, taking into account both tails of the distribution, we refer to the p-value as being two-tailed. In symmetrical distributions, two-tailed p-values may be obtained merely by doubling the onetailed probability value. Thus, the probability of selecting a person with a score as deviant as 132 is 2 X 0.0228 = 0.0456.
PROBABILITY Page 9 of 9 5. The distinction between one- and two-tailed probability values takes on added significance as we progress into inferential statistics.