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:

Step1: Recall independent - events definition

Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other event.

Step2: Analyze rolling two number cubes

The result of rolling one number cube does not affect the result of rolling the other number cube. So, rolling two number cubes at the same time is an independent - event.

Step3: Analyze choosing a card and flipping a coin

Choosing a card has no impact on the outcome of flipping a coin. So, choosing a card and then flipping a coin is an independent - event.

Step4: Analyze choosing first, second, and third - place winner

The choice of the first - place winner affects the choices for the second - place winner, and the choice of the first and second - place winners affects the choice of the third - place winner. So, this is a dependent - event.

Step5: Analyze drawing a name from a hat and then drawing another

When a name is drawn and kept, the sample space for the second draw is reduced. So, this is a dependent - event.

Step6: Analyze drawing a marble, replacing it, and drawing another

Since the marble is replaced, the sample space remains the same for the second draw. The first draw does not affect the second draw. So, this is an independent - event.

Step7: Analyze flipping the same coin three times

The result of one coin - flip does not affect the result of the other coin - flips. So, flipping the same coin three times is an independent - event.

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