In a study about a pandemic, data of 900 persons was collected. It was found that 190 persons had sy

Q: Wait, why do we subtract the central intersection (30) from the pairwise intersections (80, 90, 70)?

That is a fantastic question! When we calculate $n(F \cap C)$ using the formula $n(F) + n(C) - n(F \cup C)$, the result (80) represents everyone who has both fever and cough. This group includes the 30 people who also have the third symptom (breathing problem). If we want to find people who have *only* fever and cough, we must remove that central core of 30 people who have all three. Think of it like peeling an onion; the center is shared by all layers, so you must remove it to see the distinct layers.

Q: What does 'at most one' symptom mean? Does it include zero?

Yes, absolutely! 'At most one' means the number of symptoms $S$ satisfies $S \leq 1$. This includes the case where $S = 0$ (no symptoms) and $S = 1$ (exactly one symptom). A common trap is to only calculate those with one symptom and forget the healthy individuals who have zero symptoms. Always check the boundaries of your set!

Q: Can we solve this without a Venn diagram?

You could use algebraic set notation, but it would be significantly more prone to error. The Venn diagram is a visual representation of the Inclusion-Exclusion Principle. It forces you to account for every region systematically. In competitive exams like JEE, visualization is your best defense against 'silly mistakes' where you double-count or miss a region entirely.

Question

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 _____________.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We begin with a universal set $U$ containing $n(U) = 900$ people. We define three sets representing the symptoms: Fever ($F$), Cough ($C$), and Breathing problems ($B$).

The given totals for these sets are:
$$n(F) = 190, \quad n(C) = 220, \quad n(B) = 220$$

It is crucial to remember that these values represent the **entirety** of each circle, including the regions where symptoms overlap.

### The Inclusion-Exclusion Principle

To determine the overlaps, we utilize the Principle of Inclusion-Exclusion. We calculate the pairwise intersections as follows:

$$n(F \cap C) = n(F) + n(C) - n(F \cup C) = 190 + 220 - 330 = 80$$

$$n(C \cap B) = n(C) + n(B) - n(C \cup B) = 220 + 220 - 350 = 90$$

$$n(F \cap B) = n(F) + n(B) - n(F \cup B) = 190 + 220 - 340 = 70$$

We are given that the number of people suffering from all three symptoms is $n(F \cap C \cap B) = 30$.

### Peeling the Onion

To isolate those with **exactly** two symptoms, we must subtract the central intersection (those with all three symptoms) from our pairwise results:

*   Only Fever and Cough: $80 - 30 = 50$
*   Only Cough and Breathing: $90 - 30 = 60$
*   Only Fever and Breathing: $70 - 30 = 40$

Next, we isolate those with **exactly one** symptom by subtracting the overlapping regions from the total for each category:

*   Only Fever: $190 - (50 + 30 + 40) = 70$
*   Only Cough: $220 - (50 + 30 + 60) = 80$
*   Only Breathing: $220 - (40 + 30 + 60) = 90$

### Final Calculation

To find the number of healthy individuals (zero symptoms), we subtract the sum of all symptom-bearing individuals from the total population:

$$n(	ext{Healthy}) = 900 - (70 + 80 + 90 + 50 + 60 + 40 + 30) = 900 - 420 = 480$$

The question asks for the probability of a person having **at most one** symptom. This includes those with zero symptoms and those with exactly one symptom:

$$	ext{Favorable Outcomes} = 480 + (70 + 80 + 90) = 480 + 240 = 720$$

The final probability is calculated as:

$$P = \frac{720}{900} = \frac{72}{90} = 0.8$$

Through systematic visualization, we conclude that the probability is **0.8**.

Enter Numerical Value:

Visualized Solution

Venn Diagram Setup

Individual Symptom Counts

The Inclusion-Exclusion Principle

Fever and Cough Overlap

Other Pairwise Overlaps

The Core Intersection

Exact Pair Intersections

Exactly One Symptom

People with No Symptoms

Defining "At Most One"

Total for "At Most One"

Final Probability

The Sigma Insight: Types of Sets and Set Operations

Analyzing the Setup

The Inclusion-Exclusion Principle

Peeling the Onion

Final Calculation

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