3 newspapers- a question about Inclusion–exclusion principle

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In summary: Yes, that's correct. So, the total number of people who read at least one morning newspaper and one evening newspaper is 11% of 100,000, which is 11,000 people. And for part d, we can see that the number of people who read exactly one morning newspaper and one evening newspaper is 1% of 100,000, which is 1,000 people. Thank you for helping me understand it better, Lola! (Smile) In summary, there is a city with 100,000 people and 3 newspapers: A, B, and C. 10% read A, 30% read B, and 5% read C. 8% read A and
  • #1
lola19991
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There is a city with 100,000 people, which has 3 newspapers: A, B and C. 10% read A, 30% read B, 5% read C. 8% read A and B, 2% read A and C, 4% read B and C and only 1% read all of them.
a) How much people read only one newspaper?
b) How much people read at least two newspapers?
c) If A and C are morning newspapers and B is an evening newspaper, how much people read at least one morning newspaper and one evening newspaper?
d) How much people read one morning newspaper and one evening newspaper?
--------------
I did a&b and the answers that I got are:
a) 20,000
b) 12,000
--------------
I would like to know how to solve the other parts of the question.
 
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  • #2
lola19991 said:
There is a city with 100,000 people, which has 3 newspapers: A, B and C. 10% read A, 30% read B, 5% read C. 8% read A and B, 2% read A and C, 4% read B and C and only 1% read all of them.
a) How much people read only one newspaper?
b) How much people read at least two newspapers?
c) If A and C are morning newspapers and B is an evening newspaper, how much people read at least one morning newspaper and one evening newspaper?
d) How much people read one morning newspaper and one evening newspaper?
--------------
I did a&b and the answers that I got are:
a) 20,000
b) 12,000
--------------
I would like to know how to solve the other parts of the question.

Hey Lola! (Wave)

Can you clarify what '10% read A' means exactly?
Does it mean that '10% read at least A'? Or '10% read only A'?

Anyway, for (c) we want to know:
$$\#(\text{at least 1 morning paper} \land \text{at least 1 evening paper})
=\#\Big((A \cup C) \cap B\Big)
$$
Do you know how to calculate that (and what it means)?
Typically we draw a so called Venn Diagram to figure out something like that. (Thinking)
 
  • #3
lola19991 said:
There is a city with 100,000 people, which has 3 newspapers: A, B and C. 10% read A, 30% read B, 5% read C. 8% read A and B, 2% read A and C, 4% read B and C and only 1% read all of them.
a) How much people read only one newspaper?
b) How much people read at least two newspapers?
c) If A and C are morning newspapers and B is an evening newspaper, how much people read at least one morning newspaper and one evening newspaper?
d) How much people read one morning newspaper and one evening newspaper?
--------------
I did a&b and the answers that I got are:
a) 20,000
b) 12,000
--------------
I would like to know how to solve the other parts of the question.

How did you solve the first two?
Why is Part c any different? (A or C) and B
Why is Part d any different? Subset of the answer to Part c?

Translation Hint:
How MANY people? People are countable.
How MUCH sugar? Sugar is in countable, but it is measurable.
How MANY frogs? Countable.
How MUCH air? Measurable.
 
  • #4
I like Serena said:
Hey Lola! (Wave)

Can you clarify what '10% read A' means exactly?
Does it mean that '10% read at least A'? Or '10% read only A'?

Anyway, for (c) we want to know:
$$\#(\text{at least 1 morning paper} \land \text{at least 1 evening paper})
=\#\Big((A \cup C) \cap B\Big)
$$
Do you know how to calculate that (and what it means)?
Typically we draw a so called Venn Diagram to figure out something like that. (Thinking)

It means that 10% read at least A and I would like to know how to calculate that and what it means and I know that part d is related to part c, so I would like to understand them both.
 
  • #5
lola19991 said:
It means that 10% read at least A and I would like to know how to calculate that and what it means and I know that part d is related to part c, so I would like to understand them both.

Ok. So that means we have the following Venn Diagram.
\begin{tikzpicture}
\begin{scope}[blend group = soft light]
\fill[red!30!white] ( 90:2) circle (3);
\fill[green!30!white] (210:2) circle (3);
\fill[blue!30!white] (330:2) circle (3);
\end{scope}
\node at (90:5) {$A$};
\node at (210:5) {$B$};
\node at (330:5) {$C$};
\node at (90:3) {1\%};
\node at (210:3) {19\%};
\node at (330:3) {0\%};
\node {1\%};
\node at (30:2) {1\%};
\node at (150:2) {7\%};
\node at (270:2) {3\%};
\end{tikzpicture}
We can see that the people reading exactly 1 news paper are 1% + 19% + 0% = 20% of 100,000.
That is indeed 20,000 people. Good!

For (c) we want $(A∪C)∩B$.
That is, we look at the $A$ and $C$ combined.
And from those parts only the ones that are within $B$.
That is 7% + 1% + 3% isn't it? (Wondering)
 

FAQ: 3 newspapers- a question about Inclusion–exclusion principle

1. What is the Inclusion-Exclusion Principle?

The Inclusion-Exclusion Principle is a mathematical concept that helps determine the number of elements in a union of multiple sets. It states that the number of elements in the union of two sets is equal to the sum of the number of elements in each set, minus the number of elements that are in both sets. This principle can be extended to more than two sets as well.

2. How can the Inclusion-Exclusion Principle be applied to the scenario of 3 newspapers?

In the context of 3 newspapers, the Inclusion-Exclusion Principle can be used to determine the number of readers who read at least one of the three newspapers. The first step would be to add up the number of readers for each individual newspaper. Then, we would subtract the number of readers who read two newspapers (intersection of two sets) and add back the number of readers who read all three newspapers (intersection of three sets). This will give us the total number of readers for at least one newspaper.

3. What are the limitations of the Inclusion-Exclusion Principle?

The Inclusion-Exclusion Principle assumes that each element is either in a set or not in a set, and there are no overlapping elements. It also assumes that the elements in different sets are independent of each other. These assumptions may not always hold true in real-world scenarios, making the results of the principle less accurate.

4. Can the Inclusion-Exclusion Principle be used for more than 3 sets?

Yes, the Inclusion-Exclusion Principle can be extended to any number of sets. The formula for calculating the union of n sets is:
|A1 ∪ A2 ∪ ... ∪ An| = ∑i=1n |Ai| - ∑in |Ai ∩ Aj| + ... + (-1)n+1 |A1 ∩ A2 ∩ ... ∩ An|

5. What other applications does the Inclusion-Exclusion Principle have?

The Inclusion-Exclusion Principle has various applications in different fields such as probability, combinatorics, and set theory. It is also used in counting problems, such as finding the number of ways to arrange a group of objects. Additionally, it can be applied to solve problems related to Venn diagrams and overlapping categories.

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