222 CHAPTER 5 Discrete Probability Distributions Method 2: Using Technology Technologies can be used to find binomial probabilities. The screen displays list binomial probabilities for n = 10 and p = 0.05, as in Example 2. Notice that in each display, the probability distribution is given as a table. Example 2 showed that P122 = 0.0746, and that same result can be easily obtained from technology; see the probability of 0.0746 corresponding to 2 successes in the displays. Statdisk Minitab Excel TI-85 Plus CE Overtime Rule in Football EXAMPLE 3 In the Chapter Problem, we noted that between 1974 and 2011, there were 460 NFL football games decided in overtime, and 252 of them were won by the team that won the overtime coin toss. Is the result of 252 wins in the 460 games equivalent to random chance, or is 252 wins significantly high? We can answer that question by finding the probability of 252 wins or more in 460 games, assuming that wins and losses are equally likely. Using the notation for binomial probabilities, we have n = 460, p = 0.5, q = 0.5, and we want to find the sum of all probabilities for each value of x from 252 through 460. The formula is not practical here, because we would need to apply it 209 times—we don’t want to go there. Table A-1 (Binomial Probabilities) doesn’t apply because n = 460, which is way beyond the scope of that table. Instead, we wisely choose to use technology. The Statdisk display on the next page shows that the probability of 252 or more wins in 460 overtime games is 0.0224 (rounded), which is low (such as less than 0.05). This shows that it is unlikely that we would get 252 or more wins by chance. If we effectively rule out chance, we are left with the more reasonable explanation that the team winning the overtime coin toss has a better chance of winning the game. SOLUTION CP
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