Which events are independent?

select each correct answer.

one ball is drawn from a bag of balls after another ball is drawn without replacement.

a coin is flipped and a spinner is spun.

white marbles are drawn from a bag of marbles without replacing the marbles.

four number cubes are rolled at the same time.


i know i have asked this before but people have given me only one answer, which was c but i think i need to select more than one answer. so much for any .

Let's say we had a bag of 10 marbles in which 3 are red. The probability of pulling out a red marble is 3/10 = 0.30 = 30%

If we do not replace that red marble, then there are 3-1 = 2 red marbles left and 10-1 = 9 marbles overall leftover. The probabiity of getting another red marble is 2/9 = 0.222 = 22.2% approximately. The probability has changed. The second event probability is dependent on the first event.

If we pulled out a non-red marble on the first selection, then we would have 3 red marbles out of 9 left over so 3/9 = 1/3 = 0.333 = 33.3% is the approximate probability of getting a red marble on the second selection. We see that this is not the same as the 30% we got earlier.

The second event depends entirely on what happens during the first event, so we say the second event is dependent on the first. Or more shortly, "the two events are dependent".

So any time there is the key phrase "no replacement" it means "dependent events".

Based on what is discussed in the section above, this means that answer choices A and C are NOT independent events.

The other two answer choices do describe independent events. Why? Because the act of flipping a coin does not affect the outcome of the spinner, and vice versa. This is assuming the coin and spinner are kept separate. The probability for one item's outcome remains the same regardless of what happens to the other item's outcome. The same applies to rolling 4 number cubes (aka dice).