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

Explanation:

To determine independent events, we use the definition: two events are independent if the occurrence of one does not affect the probability of the other.

1. Rolling two number cubes at the same time: The outcome of one cube does not affect the outcome of the other. So, these are independent.
2. Choosing a card, and then flipping a coin: Choosing a card has no impact on the result of flipping a coin. These are independent.
3. Choosing a first, second, and third place winner: The choice of first place affects the available choices for second and third, so these are dependent.
4. Drawing a name from a hat, keeping it, then drawing another name: Removing a name changes the total number of names, affecting the probability of the next draw. These are dependent.
5. Drawing a marble from a bag, replacing it, then drawing another marble: Replacing the marble restores the original conditions, so the first draw does not affect the second. These are independent.
6. Flipping the same coin three times: Each flip is independent of the others, as the coin has no memory of previous flips. These are independent.

Answer:

• rolling two number cubes at the same time
• choosing a card, and then flipping a coin
• drawing a marble from a bag, replacing it, then drawing another marble
• flipping the same coin three times