A container holds 12 red marbles, 10 blue marbles, and 14 green marbles. You will randomly select two marbles without replacement.
-Fill in the probabilities on each branch of the tree diagram. Use the boxes with the fraction bars already provided.
-Use the tree diagram to answer the following:
-How many ways can you select the marbles?
-How many ways can you select exactly 1 blue marble?
-What is the probability that you select 2 red marbles?
-What is the probability that you select a green marble and then a red marble?
Thanks!
Boxes not given.
To fill in the probabilities on each branch of the tree diagram, we need to calculate the probabilities of each marble being selected at each step.
Step 1: Selecting the first marble
There are a total of 36 marbles in the container, so the probability of selecting a red marble is 12/36 = 1/3, the probability of selecting a blue marble is 10/36 = 5/18, and the probability of selecting a green marble is 14/36 = 7/18.
Step 2: Selecting the second marble
After the first marble is selected, we have two cases to consider: selecting another marble of the same color or selecting a different color.
Case 1: Selecting a marble of the same color as the first one
If the first marble selected was red, then there are 11 red marbles remaining out of a total of 35 marbles. Therefore, the probability of selecting a red marble after selecting a red marble is 11/35.
If the first marble selected was blue, then there are 9 blue marbles remaining out of a total of 35 marbles. Therefore, the probability of selecting a blue marble after selecting a blue marble is 9/35.
If the first marble selected was green, then there are 13 green marbles remaining out of a total of 35 marbles. Therefore, the probability of selecting a green marble after selecting a green marble is 13/35.
Case 2: Selecting a marble of a different color than the first one
If the first marble selected was red, then there are 24 marbles of other colors remaining out of a total of 35 marbles. Therefore, the probability of selecting a marble of a different color after selecting a red marble is 24/35.
If the first marble selected was blue, then there are 26 marbles of other colors remaining out of a total of 35 marbles. Therefore, the probability of selecting a marble of a different color after selecting a blue marble is 26/35.
If the first marble selected was green, then there are 22 marbles of other colors remaining out of a total of 35 marbles. Therefore, the probability of selecting a marble of a different color after selecting a green marble is 22/35.
Now we can fill in the probabilities on each branch of the tree diagram:
|___ 1/3 ___ 11/35 (Same color)
|
|___ 5/18 ___ 9/35 (Same color)
|
|___ 7/18 ___ 13/35 (Same color)
|
|___ 1/3 ___ 24/35 (Different color)
|
|___ 5/18 ___ 26/35 (Different color)
|
|___ 7/18 ___ 22/35 (Different color)
Now let's answer your questions using the tree diagram:
1. How many ways can you select the marbles?
By counting the number of branches in the tree diagram, we can see that there are 6 different ways to select the marbles.
2. How many ways can you select exactly 1 blue marble?
By looking at the branches where exactly 1 blue marble is selected, we can see that there are 2 different ways to do so.
3. What is the probability that you select 2 red marbles?
By looking at the branch where 2 red marbles are selected, the probability is calculated by multiplying the probabilities along that branch: (1/3) * (11/35) = 11/105.
4. What is the probability that you select a green marble and then a red marble?
By looking at the branches where a green marble is selected first and then a red marble, we can see that there are 2 different ways to do so. We can calculate the probability by adding the probabilities along those branches: (7/18) * (13/35) + (7/18) * (22/35) = 217/630.