Question

In a class of 400
students, 150
are interested in doing a statistics project, 300
are interested in doing a machine learning project and 25
are not interested in doing either of the projects. Then the number of students who are interested in doing both the projects, is

Answer 1

Given:

Total number of students, n(U) = 400

Number of students in Statistics Project, n(A) = 150

Number of students in Machine Learning Project, n(B) = 300

Number of students not in any Project, n(A $\cup$ B)' = 25

To find:

Number of students in both the projects, n(A $\cap$ B) = ?

Solution:

We know the formula:

$n(A\cup B) = n(A) +n(B)-n(A\cap B)$

To find $n(A\cap B)$ we need $n(A \cup B)$.

We know that total number of students is equal to the sum of students in at least one project and student not in any project.

i.e.

$n(U) = n(A\cup B)+n(A\cup B)'\\\Rightarrow 400 = n(A\cup B)+25\\\Rightarrow n(A\cup B)=375$

Now, putting the values in the above formula:

$375 = 150+300-n(A\cap B)\\\Rightarrow 375 = 450-n(A\cap B)\\\Rightarrow n(A\cap B)= 450-375 \\\Rightarrow n(A\cap B)= 75$

So, answer is:

Number of students in both the projects, n(A $\cap$ B) = 75

Answer 2

Step-by-step explanation:

Given:

Total number of students, n(U) = 400

Number of students in Statistics Project, n(A) = 150

Number of students in Machine Learning Project, n(B) = 300

Number of students not in any Project, n(A \cup∪ B)' = 25

To find:

Number of students in both the projects, n(A \cap∩ B) = ?

Solution:

We know the formula:

n(A\cup B) = n(A) +n(B)-n(A\cap B)n(A∪B)=n(A)+n(B)−n(A∩B)

To find n(A\cap B)n(A∩B) we need n(A \cup B)n(A∪B) .

We know that total number of students is equal to the sum of students in at least one project and student not in any project.

i.e.

\begin{gathered}n(U) = n(A\cup B)+n(A\cup B)'\\\Rightarrow 400 = n(A\cup B)+25\\\Rightarrow n(A\cup B)=375\end{gathered}

n(U)=n(A∪B)+n(A∪B)

′

⇒400=n(A∪B)+25

⇒n(A∪B)=375

Now, putting the values in the above formula:

\begin{gathered}375 = 150+300-n(A\cap B)\\\Rightarrow 375 = 450-n(A\cap B)\\\Rightarrow n(A\cap B)= 450-375 \\\Rightarrow n(A\cap B)= 75\end{gathered}

375=150+300−n(A∩B)

⇒375=450−n(A∩B)

⇒n(A∩B)=450−375

⇒n(A∩B)=75

now convert 75 in present

75/400*100 = 18.75

convert in to decimal places and round off

answer = 0.187 = 0.19