Chapter 5, Problem 66CP

Peyton Manning completions As of the end of the 2010 NFL season, Indianapolis Colts quarterback Peyton Manning, throughout his 13-year career, completed 65% of all of his pass attempts. Suppose the probability each pass attempted in the next season has probability 0.65 of being completed.

1. a. Does this mean that if we watch Manning throw 100 times in the upcoming season, he would complete exactly 65 passes? Explain.
2. b. Explain what this probability means in terms of observing him over a longer period, say for 1000 passes over the course of the next two seasons, assuming Manning is still at his typical playing level. Would it be surprising if his completion percentage over a large number of passes differed significantly from 65%?

To determine

Check whether the meaning of probability “if one watches Person PM throwing 100 times in the upcoming season, then the person would complete exactly 65 passes” is correct or not.

Explain the reason.

Explanation of Solution

The given information is that the completion percentage for the next season is 0.65.

Law of large number:

If an experiment is repeated a large number of times, then the proportion of the certain event approaches its probability.

From the law of large number, it can be concluded that proportion approaches probability. In other words, the law of large number gives approximate values, not exact values. Therefore, it cannot be concluded that there are exactly 65 passes out of 100 throws.

Therefore, the probability value does not mean that if one watches Person PM throwing 100 times in the upcoming season, then the person would complete exactly 65 passes.

To determine

Explain the meaning of the probability in terms of observing longer periods.

Check whether it is surprising if the completion percentage is significantly differed from 65%.

Explanation of Solution

From the law of large number, it can be concluded that proportion approaches probability. In other words, the law of large number gives approximate values, not exact values. Therefore, it cannot be concluded that there are exactly 65 passes out of 100 throws.

If the person passes the throw large numbers of times, then by the law of large numbers, it can be concluded that the completion percentage is closer to 0.65. If the person is at typical playing level and the completion percentage is significantly differed from 65%, then it is surprising.