Key Concepts and Formulas

• Principle of Inclusion-Exclusion for two sets: For any two events A and B, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. This formula helps calculate the probability of at least one event occurring.
• Properties of Set Intersection: The probability of the intersection of two events, $P(A \cap B)$, cannot be greater than the probability of either individual event. Mathematically, $P(A \cap B) \le P(A)$ and $P(A \cap B) \le P(B)$.
• Total Probability Constraint: The probability of the union of events cannot exceed $100\%$ (or $1$). $P(A \cup B) \le 1$.

Step-by-Step Solution

Let $H$ be the set of patients suffering from a heart ailment, and $L$ be the set of patients suffering from a lung infection. We are given the following percentages:

• Percentage of patients with heart ailment, $P(H) = 89\%$.
• Percentage of patients with lung infection, $P(L) = 98\%$.
• Percentage of patients suffering from both ailments, $P(H \cap L) = K\%$.

Our goal is to determine the possible range of values for $K$.

Step 1: Determine the Lower Bound for K

We use the Principle of Inclusion-Exclusion: $P(H \cup L) = P(H) + P(L) - P(H \cap L)$ Substituting the given values: $P(H \cup L) = 89\% + 98\% - K\%$ $P(H \cup L) = 187\% - K\%$ The percentage of patients suffering from at least one ailment, $P(H \cup L)$, cannot exceed the total percentage of patients in the hospital, which is $100\%$. Therefore, we have: $P(H \cup L) \le 100\%$ Substituting the expression for $P(H \cup L)$: $187\% - K\% \le 100\%$ Rearranging the inequality to solve for $K$: $187 - K \le 100$ $187 - 100 \le K$ $87 \le K$ This establishes that $K$ must be at least $87$.

Step 2: Determine the Upper Bound for K

The percentage of patients suffering from both ailments, $K\%$, cannot be greater than the percentage of patients suffering from either individual ailment. This is because the group suffering from both is a subset of those with heart ailments and also a subset of those with lung infections. $P(H \cap L) \le P(H) \quad \text{and} \quad P(H \cap L) \le P(L)$ Therefore, $P(H \cap L)$ must be less than or equal to the minimum of $P(H)$ and $P(L)$: $K\% \le \min(P(H), P(L))$ Substituting the given values: $K\% \le \min(89\%, 98\%)$ $K\% \le 89\%$ This establishes that $K$ cannot be greater than $89$.

Step 3: Combine the Bounds and Identify the Possible Range for K

From Step 1, we found $K \ge 87$. From Step 2, we found $K \le 89$. Combining these inequalities, the possible range for $K$ is: $87 \le K \le 89$ This means that the percentage of patients suffering from both ailments must be between $87\%$ and $89\%$, inclusive.

Step 4: Analyze the Given Options

The question asks which set $K$ cannot belong to. This means we need to find the option where all listed values fall outside the permissible range $[87, 89]$.

• (A) {80, 83, 86, 89}: The value $89$ is within the range $[87, 89]$. Thus, $K$ can belong to this set.
• (B) {84, 86, 88, 90}: The value $88$ is within the range $[87, 89]$. Thus, $K$ can belong to this set.
• (C) {79, 81, 83, 85}: All values in this set ($79, 81, 83, 85$) are less than $87$. None of these values fall within the range $[87, 89]$. Therefore, $K$ cannot belong to this set.
• (D) {84, 87, 90, 93}: The value $87$ is within the range $[87, 89]$. Thus, $K$ can belong to this set.

The set that $K$ cannot belong to is (C).

Common Mistakes & Tips

• Overlooking the $100\%$ limit: Always remember that the union of events cannot exceed the total population ($100\%$). This is crucial for establishing the lower bound of the intersection.
• Confusing Union and Intersection: Understand that $P(A \cup B)$ means "A or B or both," while $P(A \cap B)$ means "A and B."
• Careful with inequalities: Ensure all algebraic manipulations of inequalities are performed correctly, especially when multiplying or dividing by negative numbers (though not applicable here).

Summary

This problem utilizes the Principle of Inclusion-Exclusion and basic set properties to establish the possible range for the percentage of patients suffering from two ailments simultaneously. The lower bound for the intersection ($K$) is derived by ensuring the union of the two conditions does not exceed $100\%$. The upper bound is determined by the smaller of the two individual percentages. The derived range for $K$ is $[87, 89]$. Any option containing values exclusively outside this range is the correct answer.

The final answer is \boxed{C}.