Chapter 8 Additional Probability Topics 8.6 The Binomial Probability Model Sometimes experiments are simulated using a random number function instead of actually performing the experiment. In Problems 76 79, use a graphing utility to simulate each experiment. 77. Tossing a Loaded Coin Consider the experiment of tossing a loaded coin 4 times and counting the number of heads occurring in these 4 tosses. Simulate the experiment using a random number function on your calculator, considering a toss to be tails (T) if the result is less than 0.80 and considering a toss to be heads (H) if the result is greater than or equal to 0.80. Record the number of heads in 4 tosses. [Note: Most calculators repeat the action of the last entry if you simply press the ENTER, or EXE, key again.] Repeat the experiment times, obtaining a sequence of numbers. Using these numbers, you can estimate the P k, for each k = 0,1, 2, 3, 4 by the ratio probability of k heads, ( ) Number of times k appears in your sequence Enter your estimates in a table. Calculate the actual probabilities using the binomial probability formula and enter these numbers in the table. How close are your numbers to the actual values? k Your Estimate of P(k) Actual Value of P(k) 0 1 2 3 4 75
Recall from Chapter 7, we can generate random numbers using the rand command. To simulate tossing the coin four times, use rand(4), store the results in the list L1, and go to the data editor under to view the results. Remember that we are using a random number generator, so your results will most likely differ from the results shown below. One entry in the list is greater than 0.8, so our result for our first trial is 1. Return to the home screen and press Í to repeat the experiment again. This time there were no entries in the list greater than 0.8, so the result from our second trial is 0. Again, there were no entries in the list greater than 0.8, so the result from our third trial is 0. 76
Again, there were no entries greater than 0.8, so the result from our fourth trial is 0. This time there was only one entry greater than 0.8, so the result from our fifth trial is 1. This time there were no entries greater than 0.8, so the result from our sixth trial is 0. Again, there were no entries greater than 0.8, so the result from our seventh trial is 0. 77
This time there was one entry greater than 0.8, so the result from our eighth trial is 1. This time there were no entries greater than 0.8, so the result from our ninth trial is 0. This time there was one entry greater than 0.8, so the result from our tenth trial is 1. The sequence of results obtained is { 1, 0, 0, 0,1, 0, 0,1, 0,1 }. Using these results, we estimate the probabilities P( k ) for each k = 0,1, 2, 3, 4. Enter these estimates, as well as the exact values, in the table below. k Your Estimate of P(k) Actual Value of P(k) 6 0 = 0.6 0.4096 4 1 = 0.4 0.4096 0 2 = 0.0 0.1536 0 3 = 0.0 0.0256 4 = 0.0016 0 0.0 While the some of the estimates are not real close to the actual values, the biggest difference between actual and estimate is only 0.1904. 79. Tossing a Loaded Coin Consider the experiment of tossing a loaded coin 8 times and counting the number of heads occurring in these 8 tosses. Simulate the experiment using a random number function on your calculator, considering a toss 78
to be tails (T) if the result is less than 0.80 and considering a toss to be heads (H) if the result is greater than or equal to 0.80. Record the number of heads in 8 tosses. Repeat the experiment times, obtaining a sequence of numbers. Using these numbers, you can estimate the probability of 3 heads, P ( 3), by the ratio Number of times 3 appears in your sequence Calculate the actual probability using the binomial probability formula. How close is your estimate to the actual value? To simulate tossing the coin eight times, use rand(8). Store the results in the list L1, and go to the data editor under to view the results. Remember that we are using a random number generator, so your results will most likely differ from the results shown on the next page. Be sure to scroll down the list to see all the entries. There are two entries greater than 0.80, so our result for our first trial is 2. Return to the home screen and press Í to repeat the experiment again. 79
Again, there are two entries greater than 0.80, so the result from our second trial is 2. This time there was only one entry greater than 0.80, so the result from our third trial is 1. This time there were no entries greater than 0.80, so the result from our fourth trial is 0. This time there was one entry greater than 0.80, so the result from our fifth trial is 1. 80
This time there were three entries greater than 0.80, so the result from our sixth trial is 3. This time there were no entries greater than 0.8, so the result from our seventh trial is 0. This time there were two entries greater than 0.80, so the result from our eighth trial is 2. This time there were two entries greater than 0.80, so the result from our ninth trial is 2. 81
This time there was one entry greater than 0.80, so the result from our tenth trial is 1. The sequence of results obtained is { 2, 2,1, 0,1, 3, 0, 2, 2,1 }. There are two occurrences of 3 in the list, so the estimate of P ( 3) is 1 =. The exact value is P ( 3) = 0.14680064. 0.1 The estimate is fairly close to the actual value. 82
Summary No new commands were introduced in this chapter. 83
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