If E and F are events with `P(E)leP(F)` and `P(EnnF)gt0`, then

Options

A. occurrence of E implies occurrence of F
B. occurrence of F implies occurrence of E
C. non-occurrence of E implies non-occurrence of F
D. None of the above implications holds

Answer

To solve the problem, we need to analyze the given conditions and determine the validity of the provided options based on the properties of probability.

### Given:
1. \( P(E) \leq P(F) \)
2. \( P(E \cap F) > 0 \)

### We need to evaluate the options:
- A: Occurrence of E implies occurrence of F
- B: Occurrence of F implies occurrence of E
- C: Non-occurrence of E implies non-occurrence of F
- D: None of the above implications hold

### Step-by-step Solution:

**Step 1: Understanding the implications of the given probabilities.**
- Since \( P(E \cap F) > 0 \), it indicates that events E and F have some overlap; they are not mutually exclusive.
- The condition \( P(E) \leq P(F) \) suggests that the probability of event E occurring is less than or equal to the probability of event F occurring.

**Hint for Step 1:** Consider the definitions of the events and their probabilities to understand their relationship.

**Step 2: Analyze Option A - Occurrence of E implies occurrence of F.**
- This statement suggests that whenever event E occurs, event F must also occur.
- However, since \( P(E) \leq P(F) \) and there are scenarios where E occurs without F occurring (as indicated by the overlap), this statement is not necessarily true.

**Hint for Step 2:** Think about whether the occurrence of one event guarantees the occurrence of another.

**Step 3: Analyze Option B - Occurrence of F implies occurrence of E.**
- This statement suggests that whenever event F occurs, event E must also occur.
- Similar to Option A, since there are parts of F that do not overlap with E, this statement is also not necessarily true.

**Hint for Step 3:** Consider the independence of events and whether one event can occur without the other.

**Step 4: Analyze Option C - Non-occurrence of E implies non-occurrence of F.**
- This statement suggests that if event E does not occur, then event F cannot occur either.
- However, since there are scenarios where F can occur without E (as indicated by the overlap), this statement is also not necessarily true.

**Hint for Step 4:** Reflect on the relationship between the non-occurrence of events and if they can exist independently.

**Step 5: Analyze Option D - None of the above implications hold.**
- Since we have established that none of the first three options (A, B, C) are true based on the given conditions, this option must be correct.

**Final Conclusion:**
- The correct answer is **Option D: None of the above implications hold.**