An urn contains sixteen red balls and seventeen white balls. A sample of three balls is selected at random and the number of red balls observed.
Determine the probability distribution for this experiment:
Number of Red Balls/Probability
0...Probability=?
1...Probability=?
2...Probability=?
3...Probability=?
Find the sample space which is 16 red + 17 white = 33 total. 33C3 = sample space= 5456
Then do 0 red balls = C(16,0)*C(17,3)/5456
1 red ball = 16c1* 17c2/5456
... so on
To determine the probability distribution for this experiment, we need to calculate the probability of observing each possible number of red balls in the sample.
First, let's calculate the total number of ways to select 3 balls from the urn, which is given by the combination formula:
C(n, r) = n! / ((n - r)! * r!)
Where n is the total number of balls in the urn and r is the number of balls we want to select.
In this case, we have a total of 33 balls in the urn (16 red + 17 white). So, n = 33 and r = 3.
C(33, 3) = 33! / ((33 - 3)! * 3!) = 33! / (30! * 3!) = (33 * 32 * 31) / (3 * 2 * 1) = 5456
Now, let's calculate the probability of observing each possible number of red balls in the sample:
1. Probability of observing 0 red balls:
To have 0 red balls in the sample, we need to choose all 3 balls from the white balls in the urn. So, the number of ways to choose 3 white balls is given by:
C(17, 3) = 17! / ((17 - 3)! * 3!) = 17! / (14! * 3!) = (17 * 16 * 15) / (3 * 2 * 1) = 680
The probability of observing 0 red balls is the number of ways to choose 3 white balls divided by the total number of ways to select 3 balls:
Probability of 0 red balls = C(17, 3) / C(33, 3) = 680 / 5456.
2. Probability of observing 1 red ball:
To have 1 red ball in the sample, we can choose 1 red ball and 2 white balls. The number of ways to choose 1 red and 2 white balls is given by:
C(16, 1) * C(17, 2) = (16! / ((16 - 1)! * 1!)) * (17! / ((17 - 2)! * 2!)) = (16 * 17 * 17) / (2 * 1) = 680
The probability of observing 1 red ball is the number of ways to choose 1 red and 2 white balls divided by the total number of ways to select 3 balls:
Probability of 1 red ball = C(16, 1) * C(17, 2) / C(33, 3) = 680 / 5456.
3. Probability of observing 2 red balls:
To have 2 red balls in the sample, we can choose 2 red balls and 1 white ball. The number of ways to choose 2 red and 1 white balls is given by:
C(16, 2) * C(17, 1) = (16! / ((16 - 2)! * 2!)) * (17! / ((17 - 1)! * 1!)) = (16 * 15 * 17) / (2 * 1) = 680
The probability of observing 2 red balls is the number of ways to choose 2 red and 1 white balls divided by the total number of ways to select 3 balls:
Probability of 2 red balls = C(16, 2) * C(17, 1) / C(33, 3) = 680 / 5456.
4. Probability of observing 3 red balls:
To have 3 red balls in the sample, we need to choose all 3 balls from the red balls in the urn. So, the number of ways to choose 3 red balls is given by:
C(16, 3) = 16! / ((16 - 3)! * 3!) = 16! / (13! * 3!) = (16 * 15 * 14) / (3 * 2 * 1) = 560
The probability of observing 3 red balls is the number of ways to choose 3 red balls divided by the total number of ways to select 3 balls:
Probability of 3 red balls = C(16, 3) / C(33, 3) = 560 / 5456.
So, the probability distribution for this experiment is as follows:
Number of Red Balls / Probability
0 / C(17, 3) / C(33, 3)
1 / C(16, 1) * C(17, 2) / C(33, 3)
2 / C(16, 2) * C(17, 1) / C(33, 3)
3 / C(16, 3) / C(33, 3)