Question

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 bulls - eye.

Explanation:

Step1: Recall definition of independent events

Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other.

Step2: Analyze first case

Sarah picks a number less - than 5 and keeps it, then picks an odd number. Keeping the first - picked number changes the sample space for the second pick, so the events are dependent.

Step3: Analyze second case

Rolling a number cube and spinning a spinner are two separate actions. The result of rolling the number cube does not affect the result of spinning the spinner. So these events are independent.

Step4: Analyze third case

Eliza chooses a jack, replaces it, and then chooses a black card. Since the card is replaced, the sample space for the second draw is the same as the first draw. The first draw does not affect the second draw, so these events are independent.

Step5: Analyze fourth case

Hayden chooses a black pair of socks, puts them on, and then chooses another pair. Removing the first pair of socks changes the sample space for the second pick, so the events are dependent.

Step6: Analyze fifth case

Choosing a card from a deck and throwing a dart are two separate actions. The result of choosing a card does not affect the result of throwing a dart. So these events are independent.

Answer:

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; 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.