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:

Analyze the definition of independent events

Using the Independent Events and Dependent Events knowledge points

Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other. If the outcome of the first event changes the possible outcomes or probabilities of the second event, they are dependent.

Evaluate each option

Using the Independent Events and Dependent Events knowledge points

• Option 1: "Roll a number cube, and then roll the same number cube again."

The outcome of the first roll has no effect on the probabilities of the second roll. These are independent.

• Option 2: "Flip a coin, and then flip the coin a second time."

The outcome of the first flip does not affect the outcome of the second flip. These are independent.

• Option 3: "Remove a card from a deck, and then remove a second card."

Since the first card is not replaced, the composition of the deck changes, which alters the probabilities for the second draw. These are dependent.

• Option 4: "Draw a card from a deck, replace it, and then draw a second card."

Because the card is replaced, the deck returns to its original state before the second draw. The first draw does not affect the second. These are independent.

• Option 5: "Take a marble out of a bag, and then take a second marble out of the bag."

Since the first marble is not replaced, the number and distribution of marbles in the bag change, altering the probabilities for the second draw. These are dependent.

Identify the three independent options

Using the Independent Events knowledge point

The three independent options are:

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

Answer:

• Roll a number cube, and then roll the same number cube again. (Correct answer)
• Flip a coin, and then flip the coin a second time. (Correct answer)
• 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 (Correct answer)
• Take a marble out of a bag, and then take a second marble out of the bag.