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 at the same time"

The result of one cube does not affect the result of the other cube. So, it is an independent - event.

Step3: Analyze "choosing a card, and then flipping a coin"

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

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

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

Step5: Analyze "drawing a name from a hat, keeping it, then drawing another name"

Since the first name is not replaced, the probability of drawing the second name is affected by the first draw. So, it is a dependent event.

Step6: Analyze "drawing a marble from a bag, replacing it, then drawing another marble"

Since the marble is replaced, the first draw does not affect the second draw. So, it is an independent - event.

Step7: Analyze "flipping the same coin three times"

The result of one coin - flip does not affect the results of the other coin - flips. So, it 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