The diagram below shows the contents of a jar from which you select marbles at random.

An illustration of a jar of marbles is shown. Four of the marbles are labeled with an upper R, seven of the marbles are labeled with an upper B, and five of the marbles are labeled with an upper G. The key identifies upper R to represent red marbles, upper B to represent blue marbles, and upper G to represent green marbles.

What is the probability of selecting a red marble, replacing it, and then selecting a blue marble?
What is the probability of selecting a red marble, setting it aside, and then selecting a blue marble?
Are the answers to parts (a) and (b) the same? Why or why not?

How would I work this out step by step?

there are 16 marbles in all

P(R) = 4/16
P(B) = 7/16
So, P(RB) = 4/16 * 7/16
See what you can do with the rest, ok?

that's the probability with replacement

Without replacement, the number of marbles decreases, so the denominators change.

I got 7/64, what do I do now?

To determine the probability of selecting marbles from the jar, we need to know the total number of marbles and the number of marbles of each color.

From the given information, we know there are 4 red marbles (labeled R), 7 blue marbles (labeled B), and 5 green marbles (labeled G) in the jar.

(a) Probability of selecting a red marble, replacing it, and then selecting a blue marble:

Step 1: Determine the probability of selecting a red marble on the first draw.
Since there are 4 red marbles out of a total of 4 + 7 + 5 = 16 marbles, the probability of selecting a red marble on the first draw is 4/16 or 1/4.

Step 2: Determine the probability of selecting a blue marble on the second draw after replacing the red marble.
Since there are still 7 blue marbles remaining out of a total of 16 marbles, the probability of selecting a blue marble on the second draw is 7/16.

Step 3: Multiply the probabilities from steps 1 and 2 to get the overall probability.
The overall probability is (1/4) * (7/16) = 7/64.

So, the probability of selecting a red marble, replacing it, and then selecting a blue marble is 7/64.

(b) Probability of selecting a red marble, setting it aside, and then selecting a blue marble:

Step 1: Determine the probability of selecting a red marble on the first draw.
As mentioned earlier, there are 4 red marbles out of a total of 16 marbles, so the probability of selecting a red marble on the first draw is 4/16 or 1/4.

Step 2: Determine the probability of selecting a blue marble on the second draw after setting aside the red marble.
After setting aside the red marble, there are now 16 - 1 = 15 marbles left, including 7 blue marbles. Therefore, the probability of selecting a blue marble on the second draw is 7/15.

Step 3: Multiply the probabilities from steps 1 and 2 to get the overall probability.
The overall probability is (1/4) * (7/15) = 7/60.

So, the probability of selecting a red marble, setting it aside, and then selecting a blue marble is 7/60.

(c) The answers to parts (a) and (b) are not the same. This is because in part (a), after each draw, the selected marble is replaced back into the jar, while in part (b), the selected marble is set aside and not replaced. Therefore, the composition of the jar changes between the two draws, leading to different probabilities.