Answer :
Let's solve the given problem step by step.
(a) Meaning of [tex]\( n (\overline{ P \cup F \cup L }) \)[/tex]:
The notation [tex]\( n (\overline{ P \cup F \cup L }) = 20 \)[/tex] represents the number of youths who do not play any of the mentioned games (PUBG, Free Fire, and Ludo). In this context, 20 youths do not play any of these games.
(b) Venn Diagram Representation:
A visual representation in the form of a Venn diagram is helpful to understand the overlapping sets. Below is a diagram representing the given information:
```
_________
/ \
/ P / \
/___55_____/____/_ \
/ \ | \
__/P ∩ F 20\ | P ∩ F ∩ L 5 |__\ +
\ L15 / \ 10 F ∩ L/
\___________/ \_____/
F40
Outside all the circles, we have 20 people who don’t play any of the games.
```
(c) Total Number of Youths Who Participated in the Survey:
To find the total number of youths who participated in the survey, we need to use the principle of inclusion-exclusion for three sets:
Let:
- [tex]\( P \)[/tex] be the number of youths who are fond of PUBG, which is 55.
- [tex]\( F \)[/tex] be the number of youths who are fond of Free Fire, which is 40.
- [tex]\( L \)[/tex] be the number of youths who are fond of Ludo, which is 35.
- [tex]\( P \cap F \)[/tex] be the number of youths who are fond of both PUBG and Free Fire, which is 20.
- [tex]\( F \cap L \)[/tex] be the number of youths who are fond of both Free Fire and Ludo, which is 10.
- [tex]\( L \cap P \)[/tex] be the number of youths who are fond of both Ludo and PUBG, which is 15.
- [tex]\( P \cap F \cap L \)[/tex] be the number of youths who are fond of all three games, which is 5.
- [tex]\( n (\overline{ P \cup F \cup L }) \)[/tex] be the number of youths who do not play any of these games, which is 20.
Using the principle of inclusion-exclusion, the total number of youths ([tex]\(n\)[/tex]) who participated in the survey is calculated as follows:
[tex]\[ n = P + F + L - P \cap F - F \cap L - L \cap P + P \cap F \cap L + n (\overline{ P \cup F \cup L }) \][/tex]
[tex]\[ n = 55 + 40 + 35 - 20 - 10 - 15 + 5 + 20 \][/tex]
[tex]\[ n = 110 \][/tex]
So, 110 youths participated in the survey.
(d) Ratio of Youths Fond of At Most One Game to At Least One Game:
First, let's find the number of youths who are fond of at most one game.
At most one game can be counted as those who are:
- Only PUBG
- Only Free Fire
- Only Ludo
- None (already given as 20)
Let's calculate each:
- Youths only fond of PUBG: [tex]\( P \)[/tex] - [tex]\( P \cap F \)[/tex] - [tex]\( P \cap L \)[/tex] + [tex]\( P \cap F \cap L = 55 - 20 - 15 + 5 = 25 \)[/tex]
- Youths only fond of Free Fire: [tex]\( F \)[/tex] - [tex]\( P \cap F \)[/tex] - [tex]\( F \cap L \)[/tex] + [tex]\( P \cap F \cap L = 40 - 20 - 10 + 5 = 15 \)[/tex]
- Youths only fond of Ludo: [tex]\( L \)[/tex] - [tex]\( L \cap P \)[/tex] - [tex]\( F \cap L \)[/tex] + [tex]\( P \cap F \cap L = 35 - 15 - 10 + 5 = 15 \)[/tex]
Adding these together gives the number of youths fond of at most one game:
[tex]\[ \text{At most one} = 25 + 15 + 15 + 20 = 75 \][/tex]
Now, for youths fond of at least one game:
[tex]\[ \text{At least one} = \text{Total youths} - \text{n (} \overline{ P \cup F \cup L} \text{)} \][/tex]
[tex]\[ \text{At least one} = 110 - 20 = 90 \][/tex]
Therefore, the ratio of the number of youths who are fond of at most one game to who are fond of at least one game is:
[tex]\[ \text{Ratio} = \frac{\text{At most one}}{\text{At least one}} = \frac{75}{90} = \frac{5}{6} \approx 0.8333 \][/tex]
So, the calculated ratio is approximately 0.8333.
(a) Meaning of [tex]\( n (\overline{ P \cup F \cup L }) \)[/tex]:
The notation [tex]\( n (\overline{ P \cup F \cup L }) = 20 \)[/tex] represents the number of youths who do not play any of the mentioned games (PUBG, Free Fire, and Ludo). In this context, 20 youths do not play any of these games.
(b) Venn Diagram Representation:
A visual representation in the form of a Venn diagram is helpful to understand the overlapping sets. Below is a diagram representing the given information:
```
_________
/ \
/ P / \
/___55_____/____/_ \
/ \ | \
__/P ∩ F 20\ | P ∩ F ∩ L 5 |__\ +
\ L15 / \ 10 F ∩ L/
\___________/ \_____/
F40
Outside all the circles, we have 20 people who don’t play any of the games.
```
(c) Total Number of Youths Who Participated in the Survey:
To find the total number of youths who participated in the survey, we need to use the principle of inclusion-exclusion for three sets:
Let:
- [tex]\( P \)[/tex] be the number of youths who are fond of PUBG, which is 55.
- [tex]\( F \)[/tex] be the number of youths who are fond of Free Fire, which is 40.
- [tex]\( L \)[/tex] be the number of youths who are fond of Ludo, which is 35.
- [tex]\( P \cap F \)[/tex] be the number of youths who are fond of both PUBG and Free Fire, which is 20.
- [tex]\( F \cap L \)[/tex] be the number of youths who are fond of both Free Fire and Ludo, which is 10.
- [tex]\( L \cap P \)[/tex] be the number of youths who are fond of both Ludo and PUBG, which is 15.
- [tex]\( P \cap F \cap L \)[/tex] be the number of youths who are fond of all three games, which is 5.
- [tex]\( n (\overline{ P \cup F \cup L }) \)[/tex] be the number of youths who do not play any of these games, which is 20.
Using the principle of inclusion-exclusion, the total number of youths ([tex]\(n\)[/tex]) who participated in the survey is calculated as follows:
[tex]\[ n = P + F + L - P \cap F - F \cap L - L \cap P + P \cap F \cap L + n (\overline{ P \cup F \cup L }) \][/tex]
[tex]\[ n = 55 + 40 + 35 - 20 - 10 - 15 + 5 + 20 \][/tex]
[tex]\[ n = 110 \][/tex]
So, 110 youths participated in the survey.
(d) Ratio of Youths Fond of At Most One Game to At Least One Game:
First, let's find the number of youths who are fond of at most one game.
At most one game can be counted as those who are:
- Only PUBG
- Only Free Fire
- Only Ludo
- None (already given as 20)
Let's calculate each:
- Youths only fond of PUBG: [tex]\( P \)[/tex] - [tex]\( P \cap F \)[/tex] - [tex]\( P \cap L \)[/tex] + [tex]\( P \cap F \cap L = 55 - 20 - 15 + 5 = 25 \)[/tex]
- Youths only fond of Free Fire: [tex]\( F \)[/tex] - [tex]\( P \cap F \)[/tex] - [tex]\( F \cap L \)[/tex] + [tex]\( P \cap F \cap L = 40 - 20 - 10 + 5 = 15 \)[/tex]
- Youths only fond of Ludo: [tex]\( L \)[/tex] - [tex]\( L \cap P \)[/tex] - [tex]\( F \cap L \)[/tex] + [tex]\( P \cap F \cap L = 35 - 15 - 10 + 5 = 15 \)[/tex]
Adding these together gives the number of youths fond of at most one game:
[tex]\[ \text{At most one} = 25 + 15 + 15 + 20 = 75 \][/tex]
Now, for youths fond of at least one game:
[tex]\[ \text{At least one} = \text{Total youths} - \text{n (} \overline{ P \cup F \cup L} \text{)} \][/tex]
[tex]\[ \text{At least one} = 110 - 20 = 90 \][/tex]
Therefore, the ratio of the number of youths who are fond of at most one game to who are fond of at least one game is:
[tex]\[ \text{Ratio} = \frac{\text{At most one}}{\text{At least one}} = \frac{75}{90} = \frac{5}{6} \approx 0.8333 \][/tex]
So, the calculated ratio is approximately 0.8333.
