5-3 Poisson Probability Distributions 235 The time interval is a day, and playing once each day results in n = 365 games. Because there is one winning set of numbers among the 10,000 that are possible (from 0000 to 9999), the probability of a win is p = 1>10,000. With n = 365 and p = 1>10,000, the conditions n Ú 100 and np … 10 are both satisfied, so we can use the Poisson distribution as an approximation to the binomial distribution. We first need the value of m, which is found as follows: m = np = 365# 1 10,000 = 0.0365 Having found the value of m, we can proceed to find the probability for specific values of x. Because we want the probability that x is “at least 1,” we will use the clever strategy of first finding P102, the probability of no wins in 365 days. The probability of at least one win can then be found by subtracting that result from 1. We find P102 by using x = 0, m = 0.0365, and e = 2.71828, as shown here: P102 = mx # e-m x! = 0.03650 # 2.71828-0.0365 0! = 1# 0.9642 1 = 0.9642 Using the Poisson distribution as an approximation to the binomial distribution, we find that there is a 0.9642 probability of no wins, so the probability of at least one win is 1 - 0.9642 = 0.0358. If we use the binomial distribution, we get a probability of 0.0358, so the Poisson distribution works quite well here. SOLUTION YOUR TURN. Do Exercise 17 “Mega Millions Lottery: Poisson Approximation to Binomial.” Poisson Distributions Access tech supplements, videos, and data sets at www.TriolaStats.com TECH CENTER Statdisk 1. Click Analysis in the top menu. 2. Select Probability Distributions from the dropdown menu and select Poisson Distribution from the submenu. 3. Enter the value of the mean and click Evaluate. Minitab 1. Enter the values of x for which you want probabilities (such as 0, 1, 2, 3, 4, 5) in column C1. 2. Select Calc from the top menu. 3. Select Probability Distributions from the dropdown menu and Poisson from the submenu. 4. Select Probability, enter the mean, and select C1 for Input Column. 5. Click OK . StatCrunch 1. Click Stat in the top menu. 2. Select Calculators from the dropdown menu and Poisson from the submenu. 3. In the dialog box enter the value of the mean and the value of x. Select = or the desired inequality for x. 4. Click Compute. TI-83>84 Plus Calculator 1. Press F then O keys to access the DISTR (distributions) menu. 2. Select poissonpdf and press [. 3. Enter the values for mean (m) and x to complete the command poissonpdf (M, x). Press [. Tip: Select poissoncdf in Step 2 for cumulative probability. Excel 1. Enter the values of x for which you want probabilities (such as 0, 1, 2, 3, 4, 5) in column A. 2. Select cell B1, click Insert Function ƒx, select the category Statistical, select the function POISSON.DIST and click OK. 3. Enter A1 for X and then enter the value of the mean. 4. Enter 0 in the Cumulative box. 5. Click OK and the probability will appear in cell B1. 6. Copy B1 down the column to obtain the probability for each value of x listed in column A. Tip: Enter 1in Step 4 for the cumulative Poisson distribution. R R command: dpois(x, M). TIP: Use the R command ppois(x, M) for cumulative probabilities. A complete list of R statistical commands is available at TriolaStats.com
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