Q.The probability that a person visiting a zoo will see the giraffee is $0.72$, the probability that he will see the bears is $0.84$ and the probability that he will see both is $0.52$.

This problem requires the application of the Probability Addition Rule. We are asked to find the probability that a person sees either the giraffe or the bears (or both), which is $P(G \cup B)$.

1. Identify the given probabilities: $P(\text{Giraffe}) = P(G) = 0.72$, $P(\text{Bears}) = P(B) = 0.84$, and $P(\text{Both}) = P(G \cap B) = 0.52$.
2. Apply the Addition Rule: $P(G \cup B) = P(G) + P(B) - P(G \cap B)$.
3. Substitute the values: $P(G \cup B) = 0.72 + 0.84 - 0.52$.
4. Calculate the result: $P(G \cup B) = 1.56 - 0.52 = 1.04$. …

The core concept for solving problems involving the probability of "at least one" of two events occurring is the Addition Rule for Probability. This rule helps us find the probability of event A or event B (or both) happening.

When we simply add the probabilities of two events, $P(A) + P(B)$, we run into a problem if the events can occur simultaneously. The outcomes where both A and B happen (their intersection, $A \cap B$) are counted twice: once as part of event A and once as part of event B. To correct for this double-counting, we must subtract the probability of their intersection, $P(A \cap B)$, once.

Let's apply this understanding to the given problem.

1. Define the events and list the given probabilities:

   Let $G$ be the event that a person sees the giraffes.

   Let $B$ be the event that a person sees the bears.

   We are provided with the following probabilities:

   • The probability of seeing giraffes: $P(G) = 0.72$
   • The probability of seeing bears: $P(B) = 0.84$
   • The probability of seeing both giraffes and bears: $P(G \cap B) = 0.52$

2. Identify the probability to be calculated:

   The problem implicitly asks for the probability that a person sees at least one of the giraffes or bears. This corresponds to the union of the two events, $P(G \cup B)$.

3. Apply the Addition Rule:

   Using the formula $P(G \cup B) = P(G) + P(B) - P(G \cap B)$, we substitute the given values:

   $P(G \cup B) = 0.72 + 0.84 - 0.52$

4. Perform the calculation:

   First, sum the individual probabilities:

   $0.72 + 0.84 = 1.56$

   Now, subtract the probability of seeing both:

   $P(G \cup B) = 1.56 - 0.52$

   $P(G \cup B) = 1.04$

5. Analyze the result for consistency:

   A fundamental principle of probability states that the probability of any event must be a value between $0$ and $1$, inclusive ($0 \le P(E) \le 1$). Our calculated value for $P(G \cup B)$ is $1.04$, which is greater than $1$. This indicates that the initial probabilities provided in the problem statement are mathematically inconsistent. …