CHAPTER 3 PROBABILITY FUNCTIONS A rich variety of the commonly used probability functions are available for use right on screen. They provide quick and easy solutions to the computation of various probabilities that are often very difficult to perform by hand or by use of a calculator. The computations are rapid and very accurate. All you need to do is enter a few items of information and you will quickly have the answers you need. None of the probability functions will send the results to your printer. However, all of your results are saved for you as you work along. Then, when you exit the SPPC you will be given the option of reviewing your work on screen, printing it, or storing it in a disk file of your choice. FACTORIALS When you choose the factorials option you will then need to enter the number for which you wish the factorial. For example, if you enter the value of N = 12, the program will provide you with a result of N! = 479001600. Here are a few more examples. N = 8 Factorial = N! = 40320 N = 32 Factorial = N! = 2.631308369336935e+35 N = 128 Factorial = N! = 3.856204823625802e+215 PERMUTATIONS AND COMBINATIONS By choosing the option to compute permutations and combinations, you will easily obtain both. You need only enter the values of n and r to obtain the number of permutations and the number of combinations of n things taken r at a time. Here are some examples that you might want to try for yourself. n = 78 r = 13 Permutations = 1.373031150116532e+24 Combinations = 220495674290430 n = 30 r = 15 Permutations = 2.028432049317273e+20 Combinations = 155117520 BINOMIAL PROBABILITIES Binomial probabilities are easily obtained by choosing that option from the Probability Menu. You will need to enter the value of n, your sample size, and the value of r which represents the number of successes in the sample. Then enter the value of p which is the probability of a single successful outcome. Suppose, for example, that you were to throw 30 pennies (nickles, dimes, quarters or whatever) high up into the air. When they land, you want to know the probability that exactly five will show heads up. In this example, n = 30, r = 5, and since the probability of a success for a single coin flip is .5, the value of p = .5. In this example, the probability of obtaining exactly five heads is .00013 while the probability of obtaining five OR MORE heads is .99987. The program also reports the mean and variance of the binomial distribution, and in this example we find that the Mean = 15 and the Variance = 7.5. Here is one more example. n = 56 r = 11 p = 0.3000 p( x ) = 0.04538838 p( x+) = 0.95461162 Mean = 16.80000000 Variance = 11.76000000 CHI-SQUARE PROBABILITIES Once you know the value of a Chi-square statistic and the degrees of freedom associated with it, it is a simple matter to obtain the probability for that Chi-square value. Merely choose the Chi-square option in the Probability Menu and then enter the degrees of freedom and the value of Chi-square. The following are two sample results for the Chi-square procedure. df = 1 Chi-square = 3.8600 p( r ) <= 0.04945 df = 4 Chi-square = 6.7800 p( r ) <= 0.14798 NORMAL CURVE AND t PROBABILITES Probabilities associated with the normal curve and the t-distribution are combined into a single option that you may choose from the Probability Menu. When you choose that option you need only enter the degrees of freedom for the t-statistic you have obtained or the sample size if you are justified in using the normal curve (in that case, df = N). Then enter the t- or z-statistic to obtain the probability results. The program automatically gives you the area above a positive value of t, the area below a positive value of t, the area below -t and above +t (i.e., the area "beyond" -t & +t), and the area from the mean to the t-value you entered. The following are sample results using first a small and then a large sample. df = 11 t or z = 3.860000 Area above +t = 0.00133 Area below +t = 0.99867 Area beyond -t & +t = 0.00265 Area from -t to +t = 0.99735 Area from mean to t = 0.49867 df = 289 t or z = 1.960000 Area above +t = 0.02500 Area below +t = 0.97500 Area beyond -t & +t = 0.05000 Area from -t to +t = 0.95000 Area from mean to t = 0.47500 ACCURACY: The probabilities that you obtain from this procedure are always accurate to the number of decimal positions indicated in your output. Although this procedure uses an algorithm that is very accurate it is also very slow (unless you have the math chip). Lots of heavy number crunching is involved. Other procedures in this statistical package use a very fast algorithm for computing t-probabilities but their accuracy should not be trusted beyond the third decimal position. In short, we've made some trade-offs between speed and accuracy. The reason we mention this trade-off here is to let you know that this procedure is very accurate. Thus, if you are "nervous" about t-probability estimates obtained from other procedures, or you just want to make sure you have five decimals of accuracy in your t-probabilities, you can always use this procedure to get that. PROBABILITIES FOR THE F DISTRIBUTION Probability statistics are easily obtained for the F-distribution by choosing that option from the Probability Menu. Once you choose that option you will need to enter the degrees of freedom for the numerator (dfn) of your F-ratio, the degrees of freedom for the denominator (dfd) of your F-ratio, and then the F-ratio itself. The following are examples which you may wish to try on your system. dfn = 1 dfd = 11 F-ratio = 8.7200 Probability <= 0.01314 dfn = 3 dfd = 178 F-ratio = 2.5600 Probability <= 0.05654 ACCURACY: The probabilities that you obtain from this procedure are always accurate to the number of decimal positions indicated in your output. Although this procedure uses an algorithm that is very accurate it is also very slow. Lots of heavy number crunching is involved. Other procedures in this statistical package use a very fast algorithm for computing F-probabilities but their accuracy should not be trusted beyond the third decimal position. In short, we've made some trade-offs between speed and accuracy. The reason we mention this trade-off here is to let you know that this procedure is very accurate. Thus, if you are "nervous" about F-probability estimates obtained from other procedures, or you just want to make sure you have five decimals of accuracy in your F-probabilities, you can always use this procedure to get that. POISSON PROBABILITIES In this procedure Poisson probabilities are always obtained by entering the Mean, m, of the distribution and a specific value, x, that is randomly sampled from the distribution. The procedure then reports to you the probability of obtaining the value of x as a random draw and the probability of obtaining a value that large OR LARGER. Although you know the value of the Mean and Variance (the variance of a Poisson distribution is always equal to the mean), both are reported routinely. The following are output examples of the Poisson probability function. m = 35.0000 x = 11 p( x ) = 0.00000153 p( x+) = 0.99999934 Mean = 35.00000000 Variance = 35.00000000 m = 35.0000 x = 41 p( x ) = 0.03819918 p( x+) = 0.17506195 Mean = 35.00000000 Variance = 35.00000000 GEOMETRIC PROBABILITIES Geometric probabilities are quickly obtained by entering the sample size and the probability of a single successful outcome. For example, if you enter n = 11 and p = 0.5, you will obtain the results shown below. A second example is provided for you to try on your system. n = 11 p = 0.50000 p( r ) = 0.00024414 p( r+) = 0.49951172 Mean = 2.00000000 Variance = 2.00000000 n = 31 p = 0.30000 p( r ) = 0.00000473 p( r+) = 0.69998422 Mean = 3.33333333 Variance = 7.77777778 HYPERGEOMETRIC PROBABILITES Hypergeometric probabilities require a wee more input. First, you must enter the population size, N. You must then enter the number of success, r, in the population. The number of cases in your sample, n, is then entered, and you must finally enter x which is the number of successes in your sample. The following are examples which you may want to try for yourself. N = 89 Size of population r = 41 Successes in the population n = 28 Size of sample drawn from N x = 7 Number of successes found in n p( x ) = 0.00470293 Prob of getting x successes p( x+) = 0.99859423 Prob of getting x OR MORE successes Mean = 12.8989 Variance = 4.8223 N = 114 r = 47 n = 87 x = 21 p( x ) = 7.58823665e-12 p( x+) = 1.00000000 Mean = 35.8684 Variance = 5.0369 N = 88 r = 21 n = 34 x = 8 p( x ) = 0.20212660 p( x+) = 0.61977049 Mean = 8.1136 Variance = 3.8343 EXPONENTIAL PROBABILITES In this procedure exponential probabilities are always obtained by entering the Mean, m, of the distribution and a specific value, x, that is randomly sampled from the distribution. The procedure then reports to you the probability of obtaining a value that is larger or smaller than x as a random draw. Although you know the value of the Mean and Variance (the variance of an exponential distribution is always equal to the mean), both are reported routinely. The following are output examples of the exponential probability function. Mean = 27.0000 x = 13.0000 Area above x = 0.61786735 Area below x = 0.38213265 Mean = 27.00000000 Variance = 27.00000000 Mean = 37.0000 x = 41.0000 Area above x = 0.33018304 Area below x = 0.66981696 Mean = 37.00000000 Variance = 37.00000000