A box contains 9 red marbles, 6 green marbles, and 10 black marbles. A sample of 11 marbles is to be picked from the box.
How many samples contain exactly 5 green marbles or exactly 3 black marbles?
I keep getting 934992 as the answer but my homework site says that is wrong.
Never mind! I didn't subtract the number of samples with exactly 5 green and 3 black.
To find the number of samples that contain exactly 5 green marbles or exactly 3 black marbles, we need to calculate two separate probabilities and then add them together.
1) Probability of picking exactly 5 green marbles:
The number of ways to choose 5 green marbles out of 6 is given by the combination formula:
C(6, 5) = 6! / (5! * (6-5)!) = 6
The remaining 6 marbles must be chosen from the remaining black and red marbles, which gives us:
C(19, 6) = 19! / (6! * (19-6)!) = 27132
Therefore, the probability of picking exactly 5 green marbles is:
P(5 green) = (6 * 27132) / C(25, 11)
2) Probability of picking exactly 3 black marbles:
Similarly, the number of ways to choose 3 black marbles out of 10 is:
C(10, 3) = 10! / (3! * (10-3)!) = 120
The remaining 8 marbles must be chosen from the remaining red and green marbles:
C(15, 8) = 15! / (8! * (15-8)!) = 6435
Therefore, the probability of picking exactly 3 black marbles is:
P(3 black) = (120 * 6435) / C(25, 11)
Finally, to find the total probability, we add the probabilities of both scenarios:
Total Probability = P(5 green) + P(3 black)
Now, you can calculate the correct answer using these formulas.
To find the number of samples that contain exactly 5 green marbles or exactly 3 black marbles, we will use the concept of combinations.
First, let's calculate the number of samples that contain exactly 5 green marbles:
We have 6 green marbles, so we need to choose 5 out of them. The number of ways to do this is denoted by C(6, 5) which can be calculated as follows:
C(6, 5) = 6! / (5! * (6-5)!) = 6
Now let's calculate the number of samples that contain exactly 3 black marbles:
We have 10 black marbles, so we need to choose 3 out of them. The number of ways to do this is denoted by C(10, 3) which can be calculated as follows:
C(10, 3) = 10! / (3! * (10-3)!) = 10! / (3! * 7!) = (10 * 9 * 8) / (3 * 2 * 1) = 120
Now, we need to calculate the total number of samples that satisfy either condition (exactly 5 green marbles or exactly 3 black marbles). We will add the number of samples with exactly 5 green marbles and the number of samples with exactly 3 black marbles:
Total number of samples = C(6, 5) + C(10, 3) = 6 + 120 = 126
Therefore, there are 126 samples that contain exactly 5 green marbles or exactly 3 black marbles.
It seems like there was an error in your calculation. Make sure to double-check your calculations for the combinations.