204 CHAPTER 5 Discrete Probability Distributions CHAPTER OBJECTIVES Figure 5-1 provides a visual illustration of what this chapter accomplishes. When investigating the numbers of heads in two coin tosses, we can use the following two different approaches: • Use real sample data to find actual results: Collect numbers of heads in tosses of two coins, summarize them in a frequency distribution, and find the mean x and standard deviation s (as in Chapters 2 and 3). • Find probabilities for each possible outcome: Find the probability for each possible number of heads in two tosses (as in Chapter 4), then summarize the results in a table representing a probability distribution, and then find the mean m and standard deviation s. In this chapter we merge the above two approaches as in this third approach: • Create a table of what we expect to happen: Create a table describing what we expect to happen (instead of what did happen), then find the population mean m and population standard deviation s. The table at the extreme right in Figure 5-1 is a probability distribution, because it describes the distribution using probabilities instead of frequency counts. The remainder Based on the data, has the NFL overtime rule change in 2012 resulted in outcomes that are more fair? Under the old rules for overtime, does winning the coin toss become an advantage? The result of 252 wins in 460 games is a winning rate of 54.8% for the teams that won the coin toss. Is that about the same as random chance, or is 54.8% significantly greater than 50%, so that teams winning the coin toss have an advantage? The methods of this chapter can be used to answer such questions. Since 2012, there is a 55.4% winning rate for the teams that won the overtime coin toss. The overtime winning rate of 55.4% since 2012 does not appear to be significantly different from the winning rate of 54.8% before 2012, suggesting that the rule change in 2012 did not have a significant effect. (Using methods presented in Section 9-1 or Section 11-2, we can determine that the two winning rates of 54.8% and 55.4% are not significantly different.) Count numbers of heads in tosses of two coins. Collect sample data from two coin tosses, then find statistics and create graphs. Number of heads x 0 1 2 f 27 56 17 Create a theoretical model of the expected results, then find parameters. Find the probability for each possible number of heads in two coin tosses. P(0) 5 0.25 P(1) 5 0.50 P(2) 5 0.25 Chapters 2 and 3 Chapter 4 Chapters 2 and 3 Chapter 5 Chapter 4 x 5 0.9 s 5 0.7 Number of heads in two coin tosses x 0 1 2 P(x) 0.25 0.50 0.25 m5 1.0 s5 0.7 FIGURE 5-1 TABLE 5-1 NFL Games Decided in Overtime Before 2012 Since 2012 Team Won Overtime Coin Toss and Won Game 252 67 Team Won Overtime Coin Toss and Lost Game 208 54
RkJQdWJsaXNoZXIy NjM5ODQ=