JEE Advanced 2022 — Paper 1 — Question 3

Let A denote the persons having symptoms of fever. $B$ denote the persons having symptoms of cough C denote the persons having symptoms of breathing problem

Given that $\mathrm{n}(\mathrm{A})=190, \quad \mathrm{n}(\mathrm{B})=220 ,\quad \mathrm{n}(\mathrm{C})=220$

$n(A \cap B \cap C)=30$

$\mathrm{n}(\mathrm{A} \cup \mathrm{B})=330$

$\mathrm{n}(\mathrm{B} \cup \mathrm{C})=350$

$\mathrm{n}(\mathrm{C} \cup \mathrm{A})=340$

$\mathrm{n}(\mathrm{A} \cup \mathrm{B})=\mathrm{n}(\mathrm{A})+\mathrm{b}(\mathrm{B})-\mathrm{n}(\mathrm{A} \cap \mathrm{B})$

$\Rightarrow 330=190+220-\mathrm{n}(\mathrm{A} \cap \mathrm{B}) \Rightarrow \mathrm{n}(\mathrm{A} \cap \mathrm{B})=80$

Similarly $n(B \cap C)=90$ and

$n(C \cap A)=70$

If we make the Venn diagram $\mathrm{n}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C})=\mathrm{n}(\mathrm{A})+\mathrm{n}(\mathrm{B})+\mathrm{n}(\mathrm{C})-\mathrm{n}(\mathrm{A} \cap \mathrm{B})-\mathrm{n}(\mathrm{B} \cap \mathrm{C})-\mathrm{n}(\mathrm{C} \cap \mathrm{A})+\mathrm{n}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})$

$=190+220+220-80-90-70+30=420$

$\therefore$ Number of person having atmost one symptom

$=480+70+80+90=720$

$\therefore$ Probability $=\frac{720}{900}=\frac{4}{5}=0.80$