Question

In a group of $50$ people, two tests were conducted, one for diabetes and one for blood pressure. $30$ people were diagnosed with diabetes and $40$ people were diagnosed with high blood pressure. What is the minimum number of people who were having diabetes and high blood pressure?

• A. $0$
• B. $10$
• C. $20$
• D. $30$

Answer 1

The correct option is C

$20$

Explanation for the correct answer:

To find the minimum number of people who were having diabetes and high blood pressure:

Given,

$30$ people were diagnosed with diabetes.

$40$ people were diagnosed with high blood pressure.

We know,

$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$

Let,

$P\left(A\right)$ be the number of people diagnosed with diabetes, $P\left(B\right)$ be the number of people were diagnosed with high blood pressure, $P\left(A\cap B\right)$ be the minimum number of people who were having diabetes and high blood pressure and $P\left(A\cup B\right)$ be the total number of people.

So,

$\begin{array}{rcl}P\left(A\cup B\right)& =& P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)\\ 50& =& 30+40-P\left(A\cap B\right)\\ P\left(A\cap B\right)& =& 20\end{array}$

Hence, the correct option is C.