Question

practice calculating probabilities using the addition and multiplication rules. 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:

To determine independent events, we use the definition: Two events \( A \) and \( B \) are independent if the occurrence of \( A \) does not affect the probability of \( B \), i.e., \( P(B|A) = P(B) \).

1. First Option: Sarah picks a number less than 5 and keeps it, then picks an odd number. Since she does not replace the first number, the sample space for the second pick is reduced (from 10 to 9 numbers), so the first event affects the second. Not independent.

1. Second Option: Rolling a number cube and spinning a spinner. These are separate actions; the result of rolling the cube (a multiple of 2) does not affect the probability of landing on red (and vice versa). Independent.

1. Third Option: Eliza chooses a jack, replaces it, then chooses a black card. Since the card is replaced, the deck remains the same for the second pick. The probability of choosing a black card is unaffected by the first pick (choosing a jack). Independent.

1. Fourth Option: Hayden chooses a black pair of socks and puts them on, then chooses another black pair. Removing the first pair reduces the number of black pairs (and total pairs) for the second pick, so the first event affects the second. Not independent.

1. Fifth Option: Choosing a card from a deck and throwing a dart. These are unrelated actions; the result of choosing an ace does not affect the probability of the dart hitting the bull’s - eye (and vice versa). Independent.

Answer:

B. 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.
C. Two cards are randomly chosen from a standard deck. Eliza chooses a jack, replaces it, and then chooses a black card.
E. 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.