Question

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

Explanation:

Step1: Recall definition of independent events

Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other event.

Step2: Analyze "Draw a card from a deck, replace it, and then draw a second card"

Since the first - drawn card is replaced, the composition of the deck remains the same for the second draw. The first draw does not affect the second draw. So these two draws are independent events.

Step3: Analyze "Flip a coin, and then flip the coin a second time"

The result of the first coin - flip does not influence the result of the second coin - flip. The probability of getting heads or tails on the second flip is not affected by the first flip. So these two coin - flips are independent events.

Step4: Analyze "Remove a card from a deck, and then remove a second card"

When the first card is removed, the number of cards in the deck changes for the second draw, and the probabilities of drawing different cards on the second draw are affected by the first draw. So these two draws are dependent events.

Step5: Analyze "Roll a number cube, and then roll the same number cube again"

The result of the first roll of the number cube does not impact the result of the second roll. The probabilities of getting each number on the second roll remain the same regardless of the first - roll result. So these two rolls are independent events.

Step6: Analyze "Take a marble out of a bag, and then take a second marble out of the bag"

If the first marble is not replaced, the composition of the marbles in the bag changes, and the probability of drawing a particular marble on the second draw is affected by the first draw. So these two draws are dependent events.

Answer:

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