which events are independent? check all that apply. \nrolling two number cubes at the same time \nchoosing a…

which events are independent? check all that apply. \nrolling two number cubes at the same time \nchoosing a card, and then flipping a coin \nchoosing a first, second, and third place winner \ndrawing a name from a hat, keeping it, then drawing another name \ndrawing a marble from a bag, replacing it, then drawing another marble \nflipping the same coin three times

which events are independent? check all that apply. \nrolling two number cubes at the same time \nchoosing a card, and then flipping a coin \nchoosing a first, second, and third place winner \ndrawing a name from a hat, keeping it, then drawing another name \ndrawing a marble from a bag, replacing it, then drawing another marble \nflipping the same coin three times

Answer

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