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:

To determine independent events, we use the definition: two events are independent if the outcome of one does not affect the outcome of the other.

Step 1: Analyze "Roll a number cube, then roll again"

Rolling a number cube again doesn't depend on the first roll. The probability of any number on the second roll is still \( \frac{1}{6} \), regardless of the first roll. So these events are independent.

Step 2: Analyze "Flip a coin, then flip again"

Flipping a coin a second time is independent of the first flip. The probability of heads (or tails) on the second flip remains \( \frac{1}{2} \), no matter the first flip's result. So these events are independent.

Step 3: Analyze "Remove a card, then remove a second"

Removing a card without replacement changes the deck (fewer cards, different probabilities for the second draw). So the second event depends on the first. Not independent.

Step 4: Analyze "Draw a card, replace, then draw"

Replacing the card restores the deck to its original state. The second draw's probability is the same as the first, so the first draw doesn't affect the second. These events are independent.

Step 5: Analyze "Take a marble, then take a second"

Taking a marble without replacement changes the bag (fewer marbles, different probabilities for the second draw). So the second event depends on the first. Not independent.

Answer:

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