If you draw a card from a deck and do not replace it, what type of events does this represent when drawing a second card?

When you draw a card from a deck and do not replace it, the events associated with drawing a second card are **dependent events**. ### Explanation: - **Dependent Events**: These are events where the outcome or occurrence of the first event affects the outcome or occurrence of the second event. In the context of drawing cards without replacement: - **First Draw**: Suppose you draw a specific card (e.g., an Ace of Spades) from a standard 52-card deck. - **Second Draw**: Since the first card was not replaced, the deck now has 51 cards. The probability of drawing another specific card (e.g., another Ace) changes because the total number of cards and the composition of the deck have been altered by the first draw. - **Contrast with Independent Events**: If the first card were replaced before drawing the second, the events would be independent because the outcome of the first draw would not affect the second. ### Example: - **First Event**: Drawing an Ace from a deck of 52 cards. - Probability: \( \frac{4}{52} = \frac{1}{13} \) - **Second Event**: Drawing another Ace after not replacing the first. - Probability: \( \frac{3}{51} = \frac{1}{17} \) Since the probability of the second event changes based on the outcome of the first event, these are dependent events. ### Key Takeaway: When events are **dependent**, the outcome of one influences the probability of the other. Drawing cards without replacement is a classic example of dependent events in probability theory. **Answer:** They are dependent events because the outcome of the second draw relies on the first draw’s result.