A television poll was conducted among regular viewers of the national news in a certain region where the three national networks share the same time slot for the evening news. Results of the poll indicate that 40% of the viewers watch the ABC evening news, 40% watch the CBS evening news, and 20% watch the NBC evening news. Furthermore, it was found that of those viewers who watched the ABC evening news during 1 week, 75% would again watch the ABC evening news during the next week, 10% would watch the CBS news, and 15% would watch the NBC news. Of those viewers who watched the CBS evening news during 1 week, 75% would again watch the CBS evening news during the next week, 5% would watch the ABC news, and 20% would watch the NBC news. Of those viewers who watched the NBC evening news during 1 week, 85% would again watch the NBC news during the next week, 10% would watch ABC, and 5% would watch CBS. What share of the audience will each network command in the long run?
A) 53.5% would watch ABC,
20)9% would watch CBS,
25)6% would watch NBC
B) 25.6% would watch ABC,
20)9% would watch CBS,
53)5% would watch NBC
C) 20.9% would watch ABC,
25)6% would watch CBS,
53)5% would watch NBC
D) 20.9% would watch ABC,
53)5% would watch CBS,
25)6% would watch NBC
E) 25.6% would watch ABC,
53)5% would watch CBS,
20)9% would watch NBC

Question

A television poll was conducted among regular viewers of the national news in a certain region where the three national networks share the same time slot for the evening news. Results of the poll indicate that 40% of the viewers watch the ABC evening news, 40% watch the CBS evening news, and 20% watch the NBC evening news. Furthermore, it was found that of those viewers who watched the ABC evening news during 1 week, 75% would again watch the ABC evening news during the next week, 10% would watch the CBS news, and 15% would watch the NBC news. Of those viewers who watched the CBS evening news during 1 week, 75% would again watch the CBS evening news during the next week, 5% would watch the ABC news, and 20% would watch the NBC news. Of those viewers who watched the NBC evening news during 1 week, 85% would again watch the NBC news during the next week, 10% would watch ABC, and 5% would watch CBS. What share of the audience will each network command in the long run? ​
A) 53.5% would watch ABC,
20)9% would watch CBS,
25)6% would watch NBC
B) 25.6% would watch ABC,
20)9% would watch CBS,
53)5% would watch NBC
C) 20.9% would watch ABC,
25)6% would watch CBS,
53)5% would watch NBC
D) 20.9% would watch ABC,
53)5% would watch CBS,
25)6% would watch NBC
E) 25.6% would watch ABC,
53)5% would watch CBS,
20)9% would watch NBC

Quizplus · Accepted Answer

To find the command share of each network in the long run, we can use a matrix to represent the transitions from one week to the next:

|             | ABC   | CBS   | NBC   |
|-------------|-------|-------|-------|
| ABC (40%)   | 0.75  | 0.1   | 0.15  |
| CBS (40%)   | 0.05  | 0.75  | 0.2   |
| NBC (20%)   | 0.1   | 0.05  | 0.85  |

The first row of the matrix represents the probabilities of a viewer of ABC news switching to ABC, CBS, or NBC news, respectively. The same holds for the second and third rows.
To calculate the long-run probabilities, we can raise the matrix to a high power (e.g., 10 or 20) until the probabilities converge to a stable distribution. However, we can also use the fact that the long-run probabilities are the eigenvectors corresponding to the eigenvalue 1, which can be calculated using linear algebra. Note that the sum of the probabilities must be 1, so we can assume that the probability of not switching to any network is 0 at first.
Using either method, we obtain the following long-run probabilities:
- P(ABC) = 0.256
- P(CBS) = 0.209
- P(NBC) = 0.535
Therefore, the best choice is B.

A Television Poll Was Conducted Among Regular Viewers of the National