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 bull’s - eye.

Explanation:

Step1: Recall definition of independent events

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

Step2: Analyze first - case

Sarah picks a number less than 5 and keeps it, then picks an odd - number. Since she keeps the first number, the sample space for the second pick is affected. So these are dependent events.

Step3: Analyze second - case

Rolling a number cube and spinning a spinner are two separate actions. The outcome of rolling the number cube does not affect the outcome of spinning the spinner. So rolling a multiple of 2 on the number cube and landing on a red portion of the spinner are independent events.

Step4: Analyze third - case

Eliza chooses a jack, replaces it, and then chooses a black card. Since the card is replaced, the probability of choosing a black card on the second draw is not affected by the first draw. So these are independent events.

Step5: Analyze fourth - case

Hayden chooses a black pair of socks and puts them on, then chooses another black pair. Since he does not replace the first pair, the sample space for the second pick is affected. So these are dependent events.

Step6: Analyze fifth - case

Choosing a card from a standard deck and throwing a dart are two separate actions. The outcome of choosing a card does not affect the outcome of throwing a dart. So choosing an ace from the deck and hitting the bull's - eye with a dart are independent events.

Answer:

The independent events are:

• 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.