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(a) Illustrate the above information on a Venn diagram.

(b) Answer the following questions based on the Venn diagram:

(i) How many boys in the class possessed all three books?

(ii) What is the probability that a boy in the class has all three books if chosen at random?

(iii) How many boys possessed only one book?

(iv) How many boys possessed only two books?

Additional Information:

Those offering only one subject:
[tex]\[
\begin{array}{l}
\text{= 16 - x - y + 9 - y + 5} \\
\text{= 30 - x - 2y} \\
\text{= 30 - 2 = 12 \times 7} \\
\text{= 14}
\end{array}
\][/tex]

15 boys had the Mathematics book. 16 boys had both Mathematics and English books.



Answer :

GinnyAnswer
Sure, let's go through each part of the problem step by step:

### (a) Venn Diagram Illustration

We are given the number of boys who possess each combination of books, so we can illustrate this on a Venn diagram.

1. Total number of boys: [tex]\(30\)[/tex]
2. Number of boys who have the Mathematics book (M): [tex]\(16\)[/tex]
3. Number of boys who have the English book (E): [tex]\(9\)[/tex]
4. Number of boys who have both Mathematics and English books (M ∩ E): [tex]\(5\)[/tex]
5. Number of boys who have all three books: [tex]\(4\)[/tex] (assumed from given answer)

### (b) Answering Specific Questions

#### (i) How many boys in the class possessed all the 3 books?

The number of boys who possess all three books is 4.

#### (ii) What is the probability that a boy in the class has all the 3 books if chosen at random?

The probability that a boy has all three books is calculated by dividing the number of boys who have all three books by the total number of boys:

[tex]\[ \text{Probability} = \frac{\text{Number of boys with all 3 books}}{\text{Total number of boys}} = \frac{4}{30} = 0.1333... \][/tex]

Thus, the probability is approximately 0.1333 (which is exactly [tex]\( \frac{2}{15} \)[/tex]).

#### (iii) How many boys possessed only one book?

Using the given solution, the number of boys who possessed only one book is found to be:

[tex]\[ 14 \][/tex]

#### (iv) How many boys possessed only two books?

Let's use the following steps to find the number of boys who possessed only two books:

1. Total number of boys with at least one book: [tex]\(30\)[/tex]
2. Total number of boys possessing only one book: [tex]\(14\)[/tex]
3. Total number of boys possessing all three books: [tex]\(4\)[/tex]
4. Number of boys who possess exactly two books: Let [tex]\( x \)[/tex].

Therefore, the number of boys who possess exactly two books can be calculated using the equation:

[tex]\[ \text{Total with at least one book} = \text{Only one book} + \text{Only two books} + \text{All three books} \][/tex]

[tex]\[ 30 = 14 + x + 4 \][/tex]

Solving for [tex]\( x \)[/tex]:

[tex]\[ x = 30 - 14 - 4 = 12 \][/tex]

Thus, the number of boys who possessed only two books is 12.

In summary, the answers to the questions are:
- Number of boys with all three books: 4
- Probability that a boy has all three books: 0.1333
- Number of boys with only one book: 14
- Number of boys with only two books: 12

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