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Randomization in a study of TV commercials. Gonzaga University professors conducted a study of more than 1,500 television commercials and published their results in the Journal of Sociology, Social Work, and Social Welfare (Vol. 2, 2008). Commercials from eight networks鈥擜BC, FAM, FOX, MTV, ESPN, CBS, CNN, and NBC鈥攚ere sampled for 8 days, with one network randomly selected each day. The table below shows the actual order determined by random draw:

ABC鈥擩uly 6 (Wed)

FAM鈥擩uly 7 (Thr)

FOX鈥擩uly 9 (Sat)

MTV鈥擩uly 10 (Sun)

ESPN鈥擩uly 11 (Mon)

CBS鈥擩uly 12 (Tue)

CNN鈥擩uly 16 (Sat)

NBC鈥擩uly 17 (Sun)

a. What is the probability that ESPN was selected on Monday, July 11?

b. Consider the four networks chosen for the weekends (Saturday and Sunday). How many ways could the researchers select four networks from the eight for the weekend analysis of commercials? (Assume that the assignment order for the four weekend days was immaterial to the analysis.)

c. Knowing that the networks were selected at random, what is the probability that ESPN was one of the four networks selected for the weekend analysis of commercials?

Step by step solution

01

Step-by-Step SolutionStep 1: The probabilities that ESPN was chosen on Monday, July 11

The mathematical tool of likelihood is employed to look at unpredictability. It's involved with the chance (probability) of an occurrence occurring. The likelihood formula is as follows:

$\mathbf{P}\mathbf{\left(}\mathbf{E}\mathbf{\right)}\mathbf{=}\frac{\mathbf{Number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{favourable}\mathbf{}\mathbf{Outcomes}}{\mathbf{Total}\mathbf{}\mathbf{Number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{Outcomes}}$

Here, we have

$\mathbf{Total}\mathbf{}\mathbf{number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{dates}\mathbf{=}\mathbf{8}$

$\begin{array}{l}\mathbf{P}\mathbf{\left(}\mathbf{ESPN}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{8}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{125}\end{array}$

Hence, the probability that ESPN will be chosen is 0.125.

02

Identify the researchers who choose four out of eight for the weekend commercial analysis.

$$\begin{gathered} \mathbf{No}\mathbf{.}\mathbf{}\mathbf{of}\mathbf{}\mathbf{ways}\mathbf{=}{}^8\mathbf{C}_4 \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{8!}{4!脳4!} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{8脳7脳6脳5脳4脳3脳2脳1}{4脳3脳2脳1脳4脳3脳2脳1} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{40,320}{576} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{70} \end{gathered}$$

Hence, the probability is 70.

03

The probability that ESPN was one of the four networks chosen for the weekend commercial analysis

ESPN is one of the four networks; therefore, we must select 3 of the remaining 7.

$$\begin{gathered} \mathbf{Probability}\mathbf{=}\frac{{}^{\mathbf{7}}\mathbf{C}_{\mathbf{3}}}{{}^{\mathbf{8}}\mathbf{C}_{\mathbf{4}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\frac{\mathbf{7}\mathbf{!}}{\mathbf{3}\mathbf{!}\mathbf{脳}\mathbf{3}\mathbf{!}}}{\frac{\mathbf{8}\mathbf{!}}{\mathbf{4}\mathbf{!}\mathbf{脳}\mathbf{4}\mathbf{!}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\frac{\mathbf{7}\mathbf{脳}\mathbf{6}\mathbf{脳}\mathbf{5}\mathbf{脳}\mathbf{4}\mathbf{脳}\mathbf{3}\mathbf{脳}\mathbf{2}\mathbf{脳}\mathbf{1}}{\mathbf{3}\mathbf{脳}\mathbf{2}\mathbf{脳}\mathbf{1}\mathbf{脳}\mathbf{3}\mathbf{脳}\mathbf{2}\mathbf{脳}\mathbf{1}}}{\frac{\mathbf{8}\mathbf{脳}\mathbf{7}\mathbf{脳}\mathbf{6}\mathbf{脳}\mathbf{5}\mathbf{脳}\mathbf{4}\mathbf{脳}\mathbf{3}\mathbf{脳}\mathbf{2}\mathbf{脳}\mathbf{1}}{\mathbf{4}\mathbf{脳}\mathbf{3}\mathbf{脳}\mathbf{2}\mathbf{脳}\mathbf{1}\mathbf{脳}\mathbf{4}\mathbf{脳}\mathbf{3}\mathbf{脳}\mathbf{2}\mathbf{脳}\mathbf{1}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\frac{\mathbf{5}\mathbf{,}\mathbf{040}}{\mathbf{36}}}{\frac{\mathbf{40}\mathbf{,}\mathbf{320}}{\mathbf{576}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{140}}{\mathbf{70}} \end{gathered}$$

Hence, the probability is2.

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