A survey of 1000 television viewers conducted by a local television station produced the following data:
•40% watch the news at 12:00
•60% watch the news at 18:00
•50% watch the news at 23:00
•25% watch the news at 12:00 and at 18:00
•20% watch the news at 18:00 and at 23:00
•10% watch all three news broadcasts
a) what percentages of those surveyed watch at least one of these programs?
b)what percent watch none of these broadcasts?
c) what percent view the news at 12:00 and at 18:00,but not at 23:00
e)What percent watches exactly two of these shows.
To solve this problem, we will be using set theory and the principle of inclusion-exclusion.
a) To find the percentage of those surveyed who watch at least one of these programs, we need to add the percentages of people who watch each program individually, subtract the percentages of people who watch two programs, and add back the percentage of people who watch all three programs.
First, calculate the percentage of people who watch each program individually:
- Percentage who watch the news at 12:00 = 40%
- Percentage who watch the news at 18:00 = 60%
- Percentage who watch the news at 23:00 = 50%
Next, calculate the percentage of people who watch two programs:
- Percentage who watch the news at 12:00 and 18:00 = 25%
- Percentage who watch the news at 18:00 and 23:00 = 20%
- Percentage who watch the news at 12:00 and 23:00 = ?
To find the missing percentage, we can use the principle of inclusion-exclusion. According to the principle, the sum of the percentages of people who watch two programs should be:
Percentage who watch the news at 12:00 or 18:00 = Percentage who watch the news at 12:00 + Percentage who watch the news at 18:00 - Percentage who watch the news at 12:00 and 18:00
So, Percentage who watch the news at 12:00 and 23:00 = Percentage who watch the news at 12:00 + Percentage who watch the news at 18:00 - Percentage who watch the news at 12:00 and 18:00
From the given information, we can substitute the known values to solve for the missing percentage.
Now, we can find the total percentage of people who watch at least one of these programs:
Percentage who watch at least one program = Percentage who watch the news at 12:00 + Percentage who watch the news at 18:00 + Percentage who watch the news at 23:00 - Percentage who watch the news at 12:00 and 18:00 - Percentage who watch the news at 18:00 and 23:00 - Percentage who watch the news at 12:00 and 23:00 + Percentage who watch all three news broadcasts
b) To find the percentage of people who watch none of these broadcasts, we can use the principle of complement. The percentage who watch none of these broadcasts is equal to 100% minus the percentage who watch at least one program.
c) To find the percentage who view the news at 12:00 and 18:00 but not at 23:00, we can subtract the percentage who watch all three programs from the percentage who watch the news at 12:00 and 18:00.
e) To find the percentage who watch exactly two of these shows, we can sum up the percentages who watch two programs and subtract the percentage who watch all three programs.
Solving b, c, and e requires substituting the values we calculated above.
By following these steps, you can find the answers to the percentage of people who watch at least one program, watch none of these broadcasts, view the news at 12:00 and 18:00 but not at 23:00, and watch exactly two of these shows.