Answer: The first, second, and fourth.
Explanation:
1). When you roll a cube twice, the number you get on the first roll doesn’t impact the second. As a result, it’s independent.
2). When you flip a coin twice, what you get on the first flip doesn’t impact what you get on the second. As a result, it’s independent.
3). When you remove a card from a deck, then remove a second one WITHOUT replacing the first, the first card impacts the second draw. For example, when I draw the first card, my chances of drawing a two are 1/13. If I draw a two the first time, my chances of drawing a two the second time are 1/17 (3/51. The total is fifty-one since one card is missing). Since the first draw impacted the probability of the second, the events are NOT independent.
4). When you remove a card, replace it, then draw again, your chances of drawing the same card doesn’t change. For example, my odds of drawing a three are 1/13 (or 4/52). If I draw a three the first time, then put it back and draw again, my odds are still 1/13 because I put the card back in the deck. As a result, the event is independent.
5). If I take a marble from a bag, and then take a second marble, the first draw impacts the second. For example, if a bag has three blue marbles and three green marbles, my chances of getting a blue marble the first time are 1/2 (or 3/6). If I get a blue marble and choose again, WITHOUT replacing it, my chances of choosing another blue marble are 2/5 (since one marble is missing). Since the first draw impacting the probability of the second, the events are NOT independent.