An urn contains 7 red marbles labeled (1,2,3,4,5,6,7) and 5 green marbles labeled (1,2,3,4,5) four marbles are pulled out at once. what is the probability.

A. all four marbles are red

B. more of the marbles are green than red

C. both red and green marbles are present

D. two of the marbles chosen are both labeled 5

A) prob (4reds) = (7/12)(6/11)(5/10)(4/9) = 7/99

or
Prob = C(7,4)/C(12,4) = 35/495 = 7/99

B) could be GGGG, GGGR
Prob = C(5,4)/C(12,4) + C(5,3)*C(7,1)/C(12,4
= 5/495 + 70/495 = 75/495 = 5/33

C) can't have all red or all green
prob = 1 - (all red + all green)
number of all green = C(5,4) = 5
so prob = 1 - (7/99 + 5/495) = 91/99

D) both are labelled 5
prob = C(2,2)*C(10,2)/C(10,4) = 45/495 = 1/11

To calculate the probabilities, we need to find the total number of possible outcomes and the number of favorable outcomes for each scenario.

1. Total number of possible outcomes:
In this case, the number of possible outcomes is the total number of ways to choose 4 marbles out of the 12 marbles in the urn. Using the combination formula, we can calculate this as:

12 choose 4 = 12! / (4! * (12-4)!) = 495.

A. All four marbles are red:
The number of favorable outcomes is the number of ways to choose 4 red marbles out of the 7 red marbles available. Using the combination formula:

7 choose 4 = 7! / (4! * (7-4)!) = 35.

The probability is the number of favorable outcomes divided by the total number of possible outcomes:

P(A) = 35 / 495 = 7 / 99.

B. More of the marbles are green than red:
The number of favorable outcomes is the sum of the following possibilities: 3 green and 1 red, 4 green and 0 red, and 5 green and 0 red.

- 3 green and 1 red:
5 choose 3 * 7 choose 1 = (5!/3!(5-3)!) * (7!/1!(7-1)!) = 10 * 7 = 70.
- 4 green and 0 red:
5 choose 4 = 5!/4!(5-4)! = 5.
- 5 green and 0 red:
5 choose 5 = 5!/5!(5-5)! = 1.

The total number of favorable outcomes is: 70 + 5 + 1 = 76.

P(B) = 76 / 495.

C. Both red and green marbles are present:
For this scenario, we can consider all the possible outcomes except when all four marbles are the same color. So, the number of favorable outcomes is:

Total number of possible outcomes - Number of outcomes where all 4 marbles are the same color = 495 - (7 choose 4 + 5 choose 4) = 495 - (35 + 5) = 495 - 40 = 455.

P(C) = 455 / 495.

D. Two of the marbles chosen are both labeled 5:
The number of favorable outcomes is the number of ways to choose 2 marbles labeled 5 out of the 12 marbles available, multiplied by the number of ways to choose 2 marbles that are not labeled 5. Using the combination formula, we can calculate this as:

2 choose 2 * 10 choose 2 = 1 * (10! / (2! * (10-2)!)) = (10! / (2! * 8!)) = 45.

P(D) = 45 / 495.

To determine the probability for each scenario, we need to know the total number of possible outcomes and the number of favorable outcomes for each case.

Total number of possible outcomes:
We are choosing 4 marbles out of a total of 12 marbles in the urn.

Number of favorable outcomes for each case:

A. All four marbles are red:
The number of favorable outcomes is the number of ways to choose 4 marbles from the 7 red marbles. This can be calculated using combinations. The formula for combinations is nCr = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items chosen.
In this case, n = 7 (number of red marbles) and r = 4 (number of marbles we want to choose).
So, the number of favorable outcomes is 7C4 = 7! / (4! * (7-4)!) = (7*6*5*4) / (4*3*2*1) = 7*5 = 35.

B. More of the marbles are green than red:
We need to calculate the number of ways to choose 4 marbles such that more are green than red.
This can be done by finding the number of ways to choose 4 green marbles and subtracting the number of ways to choose all red marbles.
Number of ways to choose 4 green marbles is 5C4 = 5! / (4! * (5-4)!) = 5! / (4! * 1!) = 5.
Number of ways to choose all red marbles is 7C4 = 35 (calculated in case A).
So, the number of favorable outcomes is 5 - 35 = -30. Since we cannot have negative outcomes, the probability for this case is 0.

C. Both red and green marbles are present:
The number of favorable outcomes is the number of ways to choose any combination of marbles that includes at least one red and one green.
We can calculate this by subtracting the number of ways to choose all red marbles and the number of ways to choose all green marbles from the total number of outcomes.
Number of ways to choose all red marbles: 7C4 = 35 (calculated in case A).
Number of ways to choose all green marbles: 5C4 = 5 (calculated in case B).
So, the number of favorable outcomes is total outcomes - (all red outcomes + all green outcomes) = 12C4 - (7C4 + 5C4) = (12*11*10*9) / (4*3*2*1) - (7*5) = 495 - 35 = 460.

D. Two of the marbles chosen are both labeled 5:
The number of favorable outcomes is the number of ways to choose 2 marbles labeled 5 and any other 2 marbles from the remaining marbles.
Number of ways to choose 2 marbles labeled 5: 2C2 = 1 (since there are exactly 2 marbles labeled 5).
Number of ways to choose the other 2 marbles: 10C2 = 10! / (2! * (10-2)!) = 10! / (2! * 8!) = (10*9) / (2*1) = 45.
So, the number of favorable outcomes is 1 * 45 = 45.

Finally, we can calculate the probability for each case by dividing the number of favorable outcomes by the total number of outcomes.

A. Probability of all four marbles being red:
Probability = Number of favorable outcomes / Total number of outcomes = 35 / 12C4 ≈ 0.143.

B. Probability of more marbles being green than red:
Probability = Number of favorable outcomes / Total number of outcomes = 0 / 12C4 = 0.

C. Probability of both red and green marbles being present:
Probability = Number of favorable outcomes / Total number of outcomes = 460 / 12C4 ≈ 0.23.

D. Probability of two marbles labeled 5 being chosen:
Probability = Number of favorable outcomes / Total number of outcomes = 45 / 12C4 ≈ 0.095.

Therefore, the probabilities are:

A. The probability that all four marbles are red is approximately 0.143.
B. The probability that more of the marbles are green than red is 0.
C. The probability that both red and green marbles are present is approximately 0.23.
D. The probability that two of the marbles chosen are both labeled 5 is approximately 0.095.