JEE Advanced 2022 — Paper 1 — MATH-I solution

JEE Advanced2022Paper 1MATH-I

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 of 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 all three symptoms (fever, cough and breathing problem).

If a person is chosen randomly from these 900 persons, then the probability that the person has at most one symptom is _____________.

Your answer

Solution

Let $F$ , $C$ , $B$ denote the sets of people having fever, cough and breathing problem respectively. We have $n (F) = 190$ , $n (C) = 220$ , $n (B) = 220$ and $n (F \cap C \cap B) = 30$ .

By inclusion-exclusion on pairs, $n (F \cap C) = n (F) + n (C) - n (F \cup C) = 190 + 220 - 330 = 80$ . Similarly, $n (F \cap B) = 190 + 220 - 340 = 70$ and $n (C \cap B) = 220 + 220 - 350 = 90$ .

The exclusive region counts are: only $F = 190 - (80 + 70 - 30) = 70$ , only $C = 220 - (80 + 90 - 30) = 80$ , only $B = 220 - (70 + 90 - 30) = 90$ . People with no symptom $= 900 - (n (F \cup C \cup B)) = 900 - 420 = 480$ .

Persons with at most one symptom $= 70 + 80 + 90 + 480 = 720$ .

Required probability $= \frac{720}{900} = 0.8$ .

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