Question

Consider a situation in which P(X) = and P(Y) = . If P(X and Y) is = , which best describes the events?

They are independent because P(X) · P(Y) = P(X and Y).
They are independent because P(X) + P(Y) = P(X and Y).
They are dependent because P(X) · P(Y) = P(X and Y).
They are dependent because P(X) + P(Y) = P(X and Y).

Answer 1

They are independent because P(X) · P(Y) = P(X and Y).

Explanation:

Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other event. This means that the probability of both events happening is equal to the product of the probabilities of each event happening individually.

In the given situation, we have:

P(X) = 1/2

P(Y) = 1/2

P(X and Y) = 1/4

If we calculate the product of P(X) and P(Y), we get:

P(X) · P(Y) = 1/2 · 1/2 = 1/4

This is equal to P(X and Y), which means that the events X and Y are independent.

P(X) + P(Y) = P(X and Y) is not a necessary condition for independence. For example, the events of flipping a coin and rolling a die are independent, but P(X) + P(Y) ≠ P(X and Y).

Therefore, the best description of the events is that they are independent because P(X) · P(Y) = P(X and Y).