Question

Which events are independent? Check all that apply.

rolling two number cubes at the same time
choosing a card, and then flipping a coin
choosing a first, second, and third place winner
drawing a name from a hat, keeping it, then drawing another name
drawing a marble from a bag, replacing it, then drawing another marble
flipping the same coin three times

Answer 1

Independent:
- rolling two number cubes at the same time — each die roll does not affect the other.
- choosing a card, and then flipping a coin — the card outcome does not affect the coin flip.
- drawing a marble from a bag, replacing it, then drawing another marble — replacement restores original probabilities, so draws are independent.
- flipping the same coin three times — separate tosses do not affect one another.

Not independent:
- choosing a first, second, and third place winner — once a person is chosen for a place, they cannot be chosen again, so probabilities change.
- drawing a name from a hat, keeping it, then drawing another name — without replacement, the second draw depends on the first.