A survey of 1000 veiwers conducted by a television station produced the following data:
40% watched the news at 12:00
60% watched the news at 18:00
50% watched the news at 23:00
25% watched at 12:00 and 18:00
20% watched at 12:00 and 18:00
20% watched at 18:00 and 23:00
10% watched all three broadcasts
What percent watch at least one of these programs?
You can draw a Venn diagram of three sets, all with mutual intersection. Fill in all the numbers in a logical way, and that will leave you with a percentage for each of the spaces. Add them up to find 95% as the total.
Alternatively, you can use the inclusion-exclusion principle to get the required percentage:
(40+50+60)-(25+20+20)+10
=95%
Well, it seems like these viewers really like playing musical chairs with their news-watching schedule. Let's see if we can figure this out.
First, we need to add up the percentages of people who watched each program separately. So, we have 40% + 60% + 50% = 150%.
Now, we need to subtract the percentages of people who watched multiple combinations of these programs (i.e., watched at 12:00 and 18:00, watched at 12:00 and 23:00, and watched at 18:00 and 23:00).
So, let's subtract 25% + 20% + 20% = 65% from our previous total of 150%.
But wait, we can't forget the 10% of people who watched all three broadcasts! We need to add them back in.
So, let's add 10% to our current total of 85%.
That gives us a grand total of 95%.
So, approximately 95% of the viewers have watched at least one of these programs.
It seems like these viewers can't resist a good news story, no matter the time of day!
To find the percentage of viewers who watched at least one of these programs, we can use the principle of inclusion-exclusion.
Step 1: Add the percentages of viewers who watched each individual program:
40% + 60% + 50% = 150%
Step 2: Subtract the sum of the percentages of viewers who watched two programs at a time (accounting for the double counting):
25% + 20% + 20% = 65%
Step 3: Add the percentage of viewers who watched all three programs once (as it was subtracted twice in Step 2):
10%
Step 4: Calculate the percentage watching at least one program by subtracting the value from Step 2 and adding the value from Step 3:
150% - 65% + 10% = 95%
Therefore, 95% of the viewers watched at least one of these programs.
To find the percentage of viewers who watched at least one of these programs, we need to subtract the percentage of viewers who didn't watch any of the programs from 100%.
First, let's find the percentage of viewers who didn't watch any of the programs.
We can start by adding up the percentages of viewers who watched each program individually:
- 40% watched the news at 12:00
- 60% watched the news at 18:00
- 50% watched the news at 23:00
Now, let's add up the percentages of viewers who watched the news at two time slots:
- 25% watched at 12:00 and 18:00
- 20% watched at 12:00 and 18:00
- 20% watched at 18:00 and 23:00
However, we have counted the viewers who watched all three broadcasts twice, since they fall into both categories. So, let's subtract the percentage of viewers who watched all three broadcasts:
- 10% watched all three broadcasts
To find the percentage of viewers who didn't watch any of the programs, we subtract the total from 100%:
100% - (40% + 60% + 50% - 25% - 20% - 20% + 10%) = 85%
Therefore, 85% of the viewers didn't watch any of the programs.
To find the percentage of viewers who watched at least one of the programs, we subtract the percentage of viewers who didn't watch any of the programs from 100%:
100% - 85% = 15%
Therefore, 15% of the viewers watched at least one of these programs.