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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.

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Find probabilities using the addition rule

Full solution

Q. 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.

Calculate Fever Only: Calculate the number of people with exactly one symptom. We use the inclusion-exclusion principle: $\newline$ - Only fever: $190 - (330 - 220) - (340 - 220) + 30 = 40$ $\newline$ - Only cough: $220 - (330 - 190) - (350 - 220) + 30 = 40$ $\newline$ - Only breathing problem: $220 - (350 - 220) - (340 - 190) + 30 = 40$
Calculate Cough Only: Add the number of people with exactly one symptom to find the total number of people with at most one symptom: $40$ (only fever) + $40$ (only cough) + $40$ (only breathing problem) = $120$

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