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.

Answer 1

Answers:
1) Independent — Even without replacement, P(second is odd) = 1/2 whether or not the first was <5, so P(A and B)=P(A)P(B).
2) Independent — the die roll and the spinner are separate physical events, so their outcomes don't affect each other.
3) Independent — the jack is replaced, so the second draw has the same probabilities as the first.
4) Not independent — choosing a black pair first changes the composition of the drawer, so the probability the next pair is black changes.
5) Independent — drawing a card and throwing a dart are unrelated, so the events do not affect each other.