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.

2. 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. These events are independent.

3. Two cards are randomly chosen from a standard deck, Eliza chooses a jack, replaces it, and then chooses a black card. These events are independent.

5. 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. These events are independent.

Step-by-step explanation:

1. 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. These events are dependent because Sarah picked a number less than 5 and kept it. Then, the probability of picking an odd number was affected (there were fewer slips of paper to choose from).

2. 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. These events are independent because to roll a cube has nothing to do with spin a spinner. So, getting a multiple of 2 does not affect the probability of landing on a red portion of the spinner.

3. Two cards are randomly chosen from a standard deck, Eliza chooses a jack, replaces it, and then chooses a black card. These events are independent, because Eliza replaces the first card, then, the probability of getting a black card was not affected.

4. 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. These events are dependent, because Hayden puts them on the first pair of socks, then, there were fewer pairs of socks in the drawer and the probability of getting the second pair of socks was affected.

5. 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. These events are independent, because to choose a card from the standard deck does not affect the probability of hitting the bull's aye with the dart.