In a study about a pandemic, data of $900$ persons was collected. It was found that
190 persons had symptom of fever
220 persons had symptom of cough
220 persons had symptom of breathing problem,
330 persons had symptom fever or cough or both
350 persons had symptom of cough or breathing problem or both
340 persons had symptom of fever or breathing problem or both
30 persons had symptom of all (fever, cough, breathing problem)
If a person is chosen randomly from these 900 persons then the probability that the person has

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Atleast two symptoms is $\frac{18}{90}$

Atleast one symptom is $\frac{42}{90}$

Exactly one symptom is $\frac{24}{90}$

Atmost one symptom is $\frac{72}{90}$

answer is A, B, C, D.

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Detailed Solution

$\begin{array}{l} n (F) = 190; n (C) = 220; n (B) = 220 \\ n (F \cup C) = 330 \Rightarrow n (F \cap C) = 80 \\ n (C \cup B) \Rightarrow 350 \Rightarrow n (C \cap B) = 90 \\ n (F \cup B) = 340 \Rightarrow n (F \cap B) = 70 \\ n (F \cap B \cap C) = 30 \\ ∴ n (F \cup C \cup B) = 420 \Rightarrow n (\bar{F \cup C \cup B}) = 480 \end{array}$

$\begin{array}{l} (A) P (A^{t} m o s t o n e s y m p t a m) = \frac{70 + 80 + 90 + 480}{900} = 0.8 \\ (B) P (E x a c t l y o n e s y m p t e n) = \frac{240}{900} \\ (C) P (a t l e r t w o) = \frac{180}{900} \\ (D) P (a t l e a r o n e s y m l) = \frac{420}{900} \end{array}$