Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
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? Show your work.
What is the probability of selecting a red marble, setting it aside, and then selecting a blue marble? Show your work.
Are the answers to parts (a) and (b) the same? Why or why not?
(a) The probability of selecting a red marble, replacing it, and then selecting a blue marble:
There are 4 red marbles, 7 blue marbles, and 5 green marbles, for a total of 4 + 7 + 5 = 16 marbles.
The probability of selecting a red marble is the ratio of red marbles to total marbles, which is 4/16 or 1/4.
Since the red marble is replaced, there are still 16 marbles in the jar when selecting the blue marble. The probability of selecting a blue marble is the ratio of blue marbles to total marbles, which is 7/16.
The probability of both events occurring in sequence is found by multiplying the probabilities of each individual event: (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) The probability of selecting a red marble, setting it aside, and then selecting a blue marble:
When we set the red marble aside, there are now only 15 marbles left in the jar. The probability of selecting a blue marble from the 15 remaining marbles is the ratio of blue marbles to total remaining marbles, which is 7/15.
The probability of both events occurring in sequence is found by multiplying the probabilities of each individual event: (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) Are the answers to parts (a) and (b) the same? Why or why not?
The answers to parts (a) and (b) are not the same. In part (a) the probability is 7/64, and in part (b) the probability is 7/60.
They are different because the way the marbles are selected affects the probability of each event. In part (a), the red marble is replaced in the jar, which means the total number of marbles remains 16 for both draws, and each draw is independent. In part (b), the red marble is set aside and not replaced, which affects the total number of marbles remaining in the jar, making the two draws dependent on each other.