## Dependent Events

Whenever you encounter probability questions involving two or more events happening at the same time or in sequence, you must first determine if their outcomes are independent or dependent.

Two events are ** dependent** if the outcome of one event affects the probability of the other.

For example, drawing a card from a deck and not returning it is an example of a dependent event. By not returning the card, you’ve decreased the total number of cards in the deck and the number of cards of that particular value (or suit, color, etc.) by one. This affects both the “bottom” and the “top” of probability calculations. If you draw a red seven, then there are 1 fewer sevens and 1 fewer red cards, so the probability of drawing a red or a seven is decreased.

Often the terms “not returned” or “without replacement” indicate that the events are dependent.

Dependent probabilities always coincide with “and” problems, so they will always be multiplication problems.

## Example

Two cards are drawn from a deck of 52 cards that contains an equal number of green, yellow, red and blue cards. They are pulled one after the other, and the first card is not returned to the deck. What is the probability that both cards are green?