Which events are independent? Check all that apply. Each number 1 through 10 is written on a slip of paper, placed in a hat, and randomly picked. Sarah picks a number less than 5 , keeps it, and then picks an odd number. A number cube is rolled and a spinner is spun. Henry rolls a multiple of 2 and lands on a red portion of the spinner. Two cards are randomly chosen from a standard deck. Eliza chooses a jack, replaces it, and then chooses a black card. Two pairs of socks are randomly chosen from a drawer. Hayden chooses a black pair of socks, puts them on, and then chooses another black pair. A card is randomly chosen from a standard deck and a dart is randomly thrown. Olivia chooses an ace and the dart hits the bull's-eye.

To determine which events are independent, we need to analyze each scenario individually. Two events are **independent** if the occurrence of one does not affect the probability of the occurrence of the other. 1. **Sarah's Number Picks:** - **Event A:** Picking a number less than 5. - **Event B:** Picking an odd number after keeping the first number. Since Sarah keeps the first number, the total numbers available for the second pick change based on whether the first number was odd or even. This affects the probability of picking an odd number subsequently. Therefore, **Events A and B are *not* independent**. 2. **Henry's Roll and Spin:** - **Event A:** Rolling a multiple of 2 on a number cube. - **Event B:** Landing on a red portion of the spinner. Rolling a die and spinning a spinner are two separate and unrelated actions. The outcome of one does not influence the outcome of the other. Therefore, **Events A and B are independent**. 3. **Eliza's Card Choices:** - **Event A:** Choosing a jack. - **Event B:** Choosing a black card after replacing the first card. Since Eliza replaces the first card back into the deck before choosing the second, the deck's composition remains unchanged. This means the probability of choosing a black card is the same regardless of the first event. Therefore, **Events A and B are independent**. 4. **Hayden's Sock Selection:** - **Event A:** Choosing a black pair of socks. - **Event B:** Choosing another black pair after putting the first pair on. By putting on the first pair of black socks, Hayden effectively removes them from the pool of available socks. This changes the probability of picking another black pair, making the second event dependent on the first. Therefore, **Events A and B are *not* independent**. 5. **Olivia's Card and Dart:** - **Event A:** Choosing an ace from a deck. - **Event B:** The dart hitting the bull's-eye. Choosing a card and throwing a dart are two completely separate actions. The outcome of the card choice does not influence the dart throw and vice versa. Therefore, **Events A and B are independent**. **Independent Events:** - **Henry's Roll and Spin** - **Eliza's Card Choices** - **Olivia's Card and Dart** **Answer:** All except the first and fourth scenarios are independent.