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:

1. Roll a number cube, then roll again: The outcome of the first roll doesn’t affect the second roll (each roll is independent, as the cube’s state doesn’t change between rolls).
2. Flip a coin, then flip again: The result of the first coin flip (heads/tails) has no impact on the second flip (the coin’s probability remains 50/50 for each flip).
3. 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 (probability of each card remains the same).

• For “Remove a card (no replacement)” and “Take a marble (no replacement)”, the first event changes the total number of items, so the second event’s probability depends on the first (dependent events).

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.