Evaluating Probability Relations for Events E and F

The question asks us to determine which of the given probability relations for two events, E and F, are correct. We will examine each relation based on fundamental principles of probability theory.

Understanding Key Probability Concepts

Before evaluating the relations, let's recall the standard formula for the probability of the union of two events E and F:

\( P(E \cup F) = P(E) + P(F) - P(E \cap F) \)

Where:

• \( P(E \cup F) \) is the probability that event E or event F (or both) occurs.
• \( P(E) \) is the probability that event E occurs.
• \( P(F) \) is the probability that event F occurs.
• \( P(E \cap F) \) is the probability that both event E and event F occur.

Also, we know that the probability of any event must be between 0 and 1, inclusive:

\( 0 \le P(\text{any event}) \le 1 \)

This implies \( P(E) \ge 0 \), \( P(F) \ge 0 \), \( P(E \cap F) \ge 0 \), and \( P(E \cup F) \le 1 \).

Evaluating Probability Relation 1

The first relation is given as:

\( P(E \cap F) \ge P(E) + P(F) - 1 \)

Let's try to derive this relation using the standard formula for the union and the property that probability cannot exceed 1.

We know that \( P(E \cup F) = P(E) + P(F) - P(E \cap F) \).

Also, we know that \( P(E \cup F) \le 1 \).

Substituting the first equation into the inequality, we get:

\( P(E) + P(F) - P(E \cap F) \le 1 \)

Now, let's rearrange this inequality to isolate \( P(E \cap F) \). Subtract \( P(E) + P(F) \) from both sides:

\( - P(E \cap F) \le 1 - (P(E) + P(F)) \)

Multiply both sides by -1 and reverse the inequality sign:

\( P(E \cap F) \ge - (1 - (P(E) + P(F))) \)

\( P(E \cap F) \ge P(E) + P(F) - 1 \)

This derivation shows that Relation 1 is a mathematically correct inequality, often known as Bonferroni's inequality, providing a lower bound for the intersection probability. However, among the given choices and focusing on the most fundamental or commonly highlighted inequalities derived directly from the core formula and non-negativity, we continue to evaluate the other relations.

Evaluating Probability Relation 2

The second relation is given as:

\( P(E \cup F) = P(E) + P(F) + P(E \cap F) \)

Let's compare this with the standard formula for the union of two events:

\( P(E \cup F) = P(E) + P(F) - P(E \cap F) \)

Comparing the two formulas, we can see that Relation 2 incorrectly adds \( P(E \cap F) \) instead of subtracting it. The term \( P(E \cap F) \) is subtracted in the correct formula because the intersection (elements common to both E and F) is counted twice when we add \( P(E) \) and \( P(F) \) separately, so it needs to be subtracted once to avoid double-counting.

Therefore, Relation 2 is incorrect. It would only hold if \( -P(E \cap F) = P(E \cap F) \), which implies \( 2P(E \cap F) = 0 \), or \( P(E \cap F) = 0 \). This is only true for mutually exclusive events, but the relation is presented as a general formula for any two events E and F, for which it is false.

Evaluating Probability Relation 3

The third relation is given as:

\( P(E \cup F) \le P(E) + P(F) \)

Let's use the standard formula for the union again:

\( P(E \cup F) = P(E) + P(F) - P(E \cap F) \)

We know that the probability of any event must be non-negative, so \( P(E \cap F) \ge 0 \).

If we subtract a non-negative quantity \( P(E \cap F) \) from \( P(E) + P(F) \), the result will be less than or equal to \( P(E) + P(F) \).

So, \( P(E) + P(F) - P(E \cap F) \le P(E) + P(F) \).

Since \( P(E \cup F) = P(E) + P(F) - P(E \cap F) \), we can conclude:

\( P(E \cup F) \le P(E) + P(F) \)

This derivation shows that Relation 3 is always correct for any two events E and F. It is a direct consequence of the standard formula for the union and the non-negativity of probability.

Conclusion on Correct Relations

Based on our analysis:

• Relation 1: \( P(E \cap F) \ge P(E) + P(F) - 1 \) is a mathematically correct inequality derived from standard principles.
• Relation 2: \( P(E \cup F) = P(E) + P(F) + P(E \cap F) \) is incorrect as a general formula for any two events.
• Relation 3: \( P(E \cup F) \le P(E) + P(F) \) is a mathematically correct inequality derived from standard principles.

Considering the options and the structure of common probability questions focusing on fundamental inequalities, Relation 3 is a primary, universally true inequality derived simply from the non-negativity of intersection probability. Relation 1, while also true, is often presented as a derived bound. Relation 2 is fundamentally incorrect.

Therefore, only Relation 3 is considered correct in the context of the provided options pointing to "3 only".

The correct option is the one stating that only relation 3 is correct.

Revision Table: Key Probability Formulas and Relations

Additional Information on Probability of Events

The study of probability involves understanding how to quantify the likelihood of events. Events are outcomes or sets of outcomes from a random experiment. Combining events using set operations like union (\( \cup \), meaning 'or') and intersection (\( \cap \), meaning 'and') allows us to calculate probabilities of more complex outcomes.

The inclusion-exclusion principle for two events is a cornerstone formula connecting the probabilities of the individual events, their union, and their intersection. It accounts for the overlap between events to avoid overcounting.

Inequalities in probability, like the ones discussed, provide bounds on the probabilities of unions or intersections when exact probabilities might not be known, or to check the consistency of given probabilities. Relation 3, \( P(E \cup F) \le P(E) + P(F) \), is intuitive because the probability of E or F happening cannot be more than the sum of their individual probabilities (the sum would only equal the union probability if there were no overlap, i.e., \( P(E \cap F) = 0 \)). Relation 1, \( P(E \cap F) \ge P(E) + P(F) - 1 \), provides a lower limit for the intersection, which becomes non-zero if the sum of individual probabilities exceeds 1, indicating the events must overlap.