Consider the following relations for two events E and F:

1. P(E ∩ F) ≥ P(E) + P(F) - 1

2. P(E ∪ F) = P(E) + P(F) + P(E ∩ F)

3. P(E ∪ F) ≤ P(E) + P(F)

Which of the above relations is/are correct?

Given:

1. P(E ∩ F) ≥ P(E) + P(F) - 1

2. P(E ∪ F) = P(E) + P(F) + P(E ∩ F)

3. P(E ∪ F) ≤ P(E) + P(F)

Calculation:

(1) P(E ∩ F) ≥ P(E) + P(F) - 1

The above inequality is the lower bound of the intersection of two events E and F. This relation is called the Principle of Inclusion-Exclusion (for two events).

If you think of this intuitively, the left-hand side P(E ∩ F) represents the probability of both events E and F happening together. In the worst case, this can't be higher than the sum of the probabilities of E and F occurring individually (i.e., P(E) + P(F)), less the probability of at least one of them happening (which is 1). So, the relation holds true that the intersection should be greater than or equal to the sum of probabilities of two events E and F, minus 1.

(2) P(E ∪ F) = P(E) + P(F) + P(E ∩ F)

The correct formula -

P(E ∪ F) = P(E) + P(F) - P(E ∩ F)

The above equation is the formula for the probability of the union of two events E and F. The original relation stated was incorrect because the plus sign before the last term P(E ∩ F) should be a minus sign.

This formula tells us the total probability of either event E, event F, or both happening is equal to the sum of their individual probabilities minus the probability of them both occurring (since we double-counted that in the sum).

(3) P(E ∪ F) ≤ P(E) + P(F)

The above inequality is the upper bound on the union of two events E and F. Essentially, this is stating that the probability of either E happening, or F happening, or both, is naturally less than or equal to the sum of their individual probabilities.

If E,F are mutually exclusive (cannot occur at the same time), the sign becomes equality instead of inequality.

So, statement 1 and 3 are correct. Hence option (3) is correct.