Understanding the Probability Problem

The question asks for the probability of the intersection of two events, A and B, given they possess three specific properties: they are equally likely, exhaustive, and independent. We need to find the value of P(A ∩ B).

Analyzing the Properties of Events A and B

Let's break down what each property means in terms of probability:

• Equally Likely: This means the probability of event A occurring is equal to the probability of event B occurring. Mathematically, this is expressed as \(P(A) = P(B)\).
• Exhaustive: This means that the union of events A and B covers the entire sample space. In other words, at least one of the events A or B must occur. Mathematically, this is expressed as \(P(A \cup B) = 1\).
• Independent: This means that the occurrence of one event does not affect the probability of the other event occurring. Mathematically, the probability of their intersection is the product of their individual probabilities: \(P(A \cap B) = P(A) \times P(B)\).

The General Formula for Union

We also know the general formula relating the probability of the union and intersection of any two events A and B:

\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

Resolving the Properties and Finding P(A ∩ B)

We are given that A and B are equally likely, exhaustive, and independent. Let \(P(A) = P(B) = p\). According to the independence property, \(P(A \cap B) = P(A) \times P(B) = p \times p = p^2\).

According to the exhaustive property, \(P(A \cup B) = 1\). Using the general formula for the union and substituting the values based on the given properties:

\[1 = p + p - p^2\]

This simplifies to:

\[1 = 2p - p^2\]

Rearranging the terms to form a quadratic equation:

\[p^2 - 2p + 1 = 0\]

This is a perfect square trinomial, which can be factored as:

\[(p - 1)^2 = 0\]

Solving for p, we get:

\[p - 1 = 0 \implies p = 1\]

This result implies that \(P(A) = 1\) and \(P(B) = 1\). If \(P(A) = 1\) and \(P(B) = 1\), then for independent events, \(P(A \cap B) = P(A) \times P(B) = 1 \times 1 = 1\). Also, \(P(A \cup B) = P(A) + P(B) - P(A \cap B) = 1 + 1 - 1 = 1\), which is consistent with the exhaustive property.

However, this leads to \(P(A \cap B) = 1\), which is not one of the provided options except implicitly if we interpret option 2 as probability 1. Given the option '0.25' is present and is indicated as the correct answer, it suggests that the question intends a scenario where \(P(A) = P(B) = 0.5\) and the events are independent, while perhaps the 'exhaustive' property is either inconsistent in this context or there is a misunderstanding in the problem statement as a whole.

Let's assume the intended scenario aligns with the probability 0.25 result. If the events are equally likely, \(P(A) = P(B)\). If they are independent, \(P(A \cap B) = P(A) \times P(B)\). If \(P(A \cap B) = 0.25\), and \(P(A) = P(B) = p\), then:

\[p \times p = 0.25\]

\[p^2 = 0.25\]

\[p = \sqrt{0.25}\]

\[p = 0.5\]

So, \(P(A) = 0.5\) and \(P(B) = 0.5\). If \(P(A) = 0.5\), \(P(B) = 0.5\), and \(P(A \cap B) = 0.25\), let's check the union probability using the general formula:

\[P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.5 + 0.5 - 0.25 = 1 - 0.25 = 0.75\]

This result \(P(A \cup B) = 0.75\) contradicts the exhaustive property \(P(A \cup B) = 1\). Therefore, the properties as stated in the question seem contradictory for events with probability less than 1.

However, if we prioritize the equally likely and independent conditions, and the likely intended answer (0.25), the derivation \(P(A)=P(B)=0.5\) and \(P(A \cap B) = 0.25\) is the most plausible path to the given correct option.

Assuming the question designers intended \(P(A) = P(B) = 0.5\) and independence:

\[P(A \cap B) = P(A) \times P(B)\]

\[P(A \cap B) = 0.5 \times 0.5\]

\[P(A \cap B) = 0.25\]

Conclusion

Based on the properties of independent and equally likely events leading to a probability of 0.25 for their intersection, and acknowledging the likely intended meaning of the question despite the conflicting 'exhaustive' property in the context of probabilities less than 1, the probability \(P(A \cap B)\) is 0.25.

Revision Table: Key Probability Concepts

Additional Information on Event Properties

Understanding the relationships between different types of events is crucial in probability theory. Let's expand on these concepts:

• Independence vs. Mutual Exclusivity: These are distinct concepts. If two events A and B with non-zero probabilities are mutually exclusive, then \(P(A \cap B) = 0\). If they are independent, \(P(A \cap B) = P(A)P(B)\). For \(P(A) > 0\) and \(P(B) > 0\), \(P(A)P(B) > 0\). Thus, events with non-zero probabilities cannot be both mutually exclusive and independent.
• Exhaustive and Mutually Exclusive Events: If a set of events is both exhaustive and mutually exclusive, they form a partition of the sample space. For two such events A and B, \(P(A \cap B) = 0\) and \(P(A \cup B) = 1\). If they are also equally likely, \(P(A) = P(B)\). Using \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\), we get \(1 = P(A) + P(A) - 0\), so \(1 = 2P(A)\), which means \(P(A) = 0.5\). In this specific case, \(P(A) = P(B) = 0.5\) and \(P(A \cap B) = 0\).
• Exhaustive and Independent Events: As shown in the problem analysis, if two events A and B with non-zero probabilities are exhaustive and independent, it implies \(P(A) = P(B) = 1\). This is because the requirement \(P(A \cup B) = 1\) combined with independence \(P(A \cup B) = P(A) + P(B) - P(A)P(B)\) and equally likely \(P(A)=P(B)=p\) leads to \(p^2 - 2p + 1 = 0\), solving to \(p=1\).

The question as posed contains a combination of properties that are contradictory unless the events have probability 1. However, standard probability problems often simplify conditions, and given the options, the most probable interpretation is that the intended scenario involves equally likely events with probability 0.5 and independence, leading to the intersection probability of 0.25.