5.2 ESTIMATING PROBABILITIES
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1 5.2 ESTIMATING PROBABILITIES It seems clear that the five-step approach of estimating expected values in Chapter 4 should also work here in Chapter 5 for estimating probabilities. Consider the following example. Note the labels for steps and 5 are changed because the goal is now to obtain an experimental probability. Example 5. In a family of three children, what is the probability that all three are boys? (Assume that boys and girls have an equal chance of being born.) One approach to solving this problem would be to take a survey of a large number of families with three children and find out in what proportion of them all three children are boys. However, it would take a lot of time to collect enough data to provide a good estimate of the probability. Another approach would be to simulate the survey by tossing three coins in succession, using the following representation: Tails (T) represent boy is born. Heads (H) represent girl is born. Note that this simulation model assumes an equal chance of a boy or a girl being born. This is approximately true in reality: about 5% of newborns are male. Suppose now that we toss three coins and get H T H. This is interpreted as follows: the first child is a girl, the second child is a boy, and the third child is a girl. We are assuming not only that boys and girls each have the same chance of being born but also that each succeeding child has an equal chance of being a boy or a girl, regardless of the sex of the previous child. We can use the five-step procedure for estimating the theoretical probability that in a family of three children, all are boys:. Choice of a Model: 2. Definition of a : Use a coin to simulate the sex of a child, where Heads: girl Tails: boy A trial consists of tossing the coin three times.. Definition of a Successful : A successful trial occurs when all three coin tosses fall tails (that is, all three children are boys). Note that from the viewpoint of the five-step method, when we are estimating the theoretical probability of an event, observing whether or not that event has occurred in the trial corresponds to observing the statistic of interest in step of Chapter Repetition of s: Do a sufficiently large number of trials (at least 00 is recommended for accuracy). Consider a sample of three trials. Instead of actually tossing coins, we will use Table B., the coin-toss random number table, which uses the digits 0 and instead of the symbols T and H. The simulated coin tosses are
2 Table 5.2 Sample s to Estimate Probability of Three Boys in a Three-Child Family THH boy No 2 HTH boy No TTT boys Yes from row of Table B., reproduced here: The three sample trials are recorded in Table Finding the Probability of a Successful : probability that all three children are boys is number of successful trials P(three boys in a family of three) total number of trials Based on only three trials, our estimate for the theoretical probability of three boys in a family of three children is P (three boys in a family of three) 0. To obtain an accurate estimate for this probability, more trials are needed. As an exercise, do 00 trials using three coins, Table B., or a computer coin-toss simulation program if you have one. If you use a computer, you can do 000 or even 0,000 trials in order to achieve extreme accuracy. The true theoretical probability is here. The estimated (experimental) How many trials should one carry out? Above we recommended that 00 trials be performed if accuracy is desired. According to advanced theory, 2 for one hundred trials typically indeed, of the time the estimated probability will be within 0.05 of the true theoretical probability. If one uses a computer to do the trials, as is possible with the instructional software provided with this textbook, then one can do a large number of trials. For example, doing 400 trials means that one will typically be within of the true theoretical probability. Why this is so will become clear in Chapter. By doing only 25 trials, one will typically be within 0. of the true probability a crude level of accuracy sometimes useful for instructional purposes. Sometimes in our examples we will demonstrate the five-step method using many trials to show the high accuracy that is possible; at other times we will use 00, 25, or fewer trials so that the approach is made clear. One always has the option to do more, even without access to computer simulation.
3 If in step we were to assign atoasuccessful trial and a0to an unsuccessful trial (this in effect being our definition of the statistic of interest for the five-step method, making the third step above the same as that in Chapter 4), then step 5 merely becomes, as it was in Chapter 4, the computation of the average of the statistic of interest. This average is simply the proportion of successes (convince yourself!). This remark makes clear that the five-step method for finding a probability is the same as the five-step method of Chapter 4. Indeed our instructional software uses the same approach for estimating a probability as for estimating an expected value. We now consider another example involving three-child families. You might have guessed (not a good method in general, because our guesses about probabilities are wrong surprisingly often) or even supplied some reasoning to convince yourself that the true probability above was the answer in the following example is less clear., but Example 5.2 In a family of three children, what is the probability that at least two are boys? (Assume that boys and girls have an equal chance of being born.) In order to solve this problem, we make only a slight modification of our five-step procedure of Example 5... Choice of a Model: Again the coin models the sex of a child: Heads: girl () Tails: boy (0) 2. Definition of a :. Definition of a Successful : A successful trial occurs when at least two of the three coin tosses fall tails (that is, at least two children are boys). This is the only change from Example Repetition of s: Do a sufficiently large number of trials, say, 00. Here we consider a sample of four trials. The coin tosses are from Table B., row. Start where we left off in Example 5. (start with the 0th character). 5. Finding the Probability of a Successful : at least two children are boys is Again, a trial consists of tossing the coin three times These four trials result in Table 5.. Number of successful trials Total number of trials The estimated probability that
4 Table 5. Sample s to Estimate Probability of At Least Two Boys in a Three-Child Family TTT boys Yes 2 THT 2 boys Yes HTH boy No 4 HHT boy No Based on our four trials, we estimate the probability of there being at least two boys in a family of three children as 2 P (at least two boys in a family of three) The true theoretical probability is 0.5; we were rather lucky that the experimental probability was exactly correct! We now consider another example from everyday life, one that does not use a fair coin model for step. Example 5. Suppose the first traffic light on your route to class is green for 20 seconds and red for 40 seconds. Thus, the light has a 60-second cycle. What is the probability that you will get exactly three green lights on the next four mornings? We need to find a model to represent the probability of of the lights being green (that is, 20/60 is the theoretical probability of the lights being green). A coin will not do the job. We could, for example, use a die having six faces. Our five-step procedure for estimating this probability is then as follows.. Choice of a Model: light (G). That is, Let one or two dots appearing on the die represent a green or 2 dots: G Let three to six dots represent no green light (N). That is,, 4, 5, or 6 dots: N 2. Definition of a : A trial consists of throwing four dice, one for each morning. (We could also use only one die and throw it four times, once for each morning.)
5 Table 5.4 Estimating the Probability of Three Green Lights on Four Mornings 246 GNNG No NGGG Yes 26 NGGN No 4 4 NNGN No. Definition of a Successful : A successful trial occurs when either one or two dots appear on exactly three of the four dice. 4. Repetition of s: Do at least 00 trials. For an example, we consider four trials. The dice rolls are from Table B.2, row. It is not good practice to always start with the first random number in row. Hence, say, start with the th digit Our four sample trials yield the results presented in Table Finding the Probability of a Successful : Based on only four trials, our estimate of the theoretical probability of getting three green lights on four mornings is number of successful trials P(getting three green lights on four mornings) total number of trials In Chapter and Chapter 4 we will learn how to solve for theoretical probabilities involving the random number of successes for a fixed number of subtrials, as in Example 5.. Such probabilities are called binomial probabilities. Examples 5. and 5.2 are also binomial probability problems. In Example 5. the theoretical probability is p (three green lights in four trials) 0. Our experimental value of 0.25 is rather far from this theoretical value. The discrepancy confirms our suspicion that four trials are likely to produce a poor estimate of the true theoretical probability.
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