Question

which events are independent? select three options. roll a number cube, and then roll the same number cube again. flip a coin, and then flip the coin a second time. remove a card from a deck, and then remove a second card. draw a card from a deck, replace it, and then draw a second card. take a marble out of a bag, and then take a second marble out of the bag.

Explanation:

• Roll a number cube, then roll again: The outcome of the first roll doesn't affect the second roll (e.g., rolling a 3 first doesn't change the probability of rolling a 5 next).
• Flip a coin, then flip again: The result of the first coin flip (heads/tails) has no impact on the second flip's probability.
• Draw a card, replace it, then draw again: Replacing the card restores the deck to its original state, so the first draw doesn't affect the second draw's probability.

• Remove a card (no replacement): Removing a card changes the deck’s composition, so the second draw’s probability depends on the first.
• Take a marble (no replacement): Removing a marble changes the bag’s contents, so the second marble’s probability depends on the first.

Answer:

A. Roll a number cube, and then roll the same number cube again.
B. Flip a coin, and then flip the coin a second time.
D. Draw a card from a deck, replace it, and then draw a second card.