Question

When drawing two hearts from a deck without replacement, are the events independent?

Answer 1

They are not independent - the first draw influences the second draw.

Explanation:

I'm going to assume that we're dealing with a standard 52-card deck (this is for ease of the explanation - the answer will be the same regardless of the makeup of the deck).

On the first draw, there are 13 hearts in the deck. The chances of drawing a heart are therefore `13/52` (which reduces to `1/4`.

If we draw another card without replacing the first card, what are the chances of drawing a second heart? There are now 12 hearts in the deck out of 51 cards total and so the odds are `12/51` (which reduces to `4/17`).

And so pulling the first card influences the odds of the second draw.

For an event to be independent, it needs to be uninfluenced by other trials. Because the draw is influenced, the second draw is not independent.