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 event: Sarah picks a number < 5 (keeps it) then picks odd. Since she keeps the first number, the sample space for the second pick is reduced (dependent, as the first pick affects the second).
2. Second event: Henry rolls a cube and spins a spinner. Rolling a cube and spinning a spinner are independent actions (the result of one does not affect the other). So "rolling a multiple of 2" and "landing on red" are independent.
3. Third event: Eliza chooses a jack (replaces it) then a black card. Replacing the card means the sample space for the second pick is unchanged (independent, as the first pick does not affect the second).
4. Fourth event: Hayden chooses a black sock (puts it on) then another black sock. Removing the first sock reduces the number of black socks (dependent, as the first pick affects the second).
5. Fifth event: Olivia chooses a card and throws a dart. These are independent actions (the card choice does not affect the dart throw, and vice versa). So "choosing an ace" and "dart hits bull’s - eye" are 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.