Nine apples, four of which are rotten, are in a refrigerator. Three apples are randomly selected without replacement. Let the random variable x represent the number chosen that are rotten. Construct a table describing the probability distribution, then find the mean and standard deviation for the random variable x.
question: WHAT STEPS WERE USED TO GET THE ANSWER?
this is the table
X P(x)
0 0.167
1 0.500
2 0.300
3 0.033
The total number of outcomes is 9C3, because there are nine apples and you are choosing 3 where order doesn't matter.
P(0) = 5C3 / 9C3
because there are 5 good apples and you need to choose 3.
P(1) = 5C2*4C1 / 9C3
P(2) = 5C1*4C2 / 9C3
P(3) = 5C0*4C3 / 9C3
THANK YOU SO MUCH I have been stumped on this question for hours on two different occasions, I am so glad you cleared this up for me.
To construct the probability distribution table for the random variable x, which represents the number of rotten apples chosen out of three, you can follow these steps:
Step 1: Determine the number of ways to choose x rotten apples out of 4 rotten apples. This can be calculated using combinations:
- When x = 0 (no rotten apples chosen):
C(4, 0) = 1
- When x = 1 (one rotten apple chosen):
C(4, 1) = 4
- When x = 2 (two rotten apples chosen):
C(4, 2) = 6
- When x = 3 (all three rotten apples chosen):
C(4, 3) = 4
Step 2: Determine the number of ways to choose (3 - x) fresh apples out of the 5 remaining fresh apples.
- When x = 0 (no rotten apples chosen):
C(5, 3) = 10
- When x = 1 (one rotten apple chosen):
C(5, 2) = 10
- When x = 2 (two rotten apples chosen):
C(5, 1) = 5
- When x = 3 (all three rotten apples chosen):
C(5, 0) = 1
Step 3: Calculate the probability of each outcome by dividing the number of favorable outcomes by the total number of possible outcomes.
- When x = 0 (no rotten apples chosen):
P(x = 0) = (1 * 10) / C(9, 3) = 0.167
- When x = 1 (one rotten apple chosen):
P(x = 1) = (4 * 10) / C(9, 3) = 0.500
- When x = 2 (two rotten apples chosen):
P(x = 2) = (6 * 5) / C(9, 3) = 0.300
- When x = 3 (all three rotten apples chosen):
P(x = 3) = (4 * 1) / C(9, 3) = 0.033
So the probability distribution table is as follows:
X P(x)
----------------
0 0.167
1 0.500
2 0.300
3 0.033
To find the mean and standard deviation for the random variable x, follow these steps:
Step 1: Multiply each value of x by its corresponding probability and sum them up to find the expected value or mean:
Mean (μ) = ∑ (x * P(x))
Mean (μ) = (0 * 0.167) + (1 * 0.500) + (2 * 0.300) + (3 * 0.033)
Mean (μ) = 0 + 0.500 + 0.600 + 0.099
Mean (μ) = 1.199
So the mean of x is 1.199.
Step 2: Calculate the variance by finding the sum of each squared difference between each value of x and the mean, multiplied by their corresponding probabilities:
Variance (σ^2) = ∑((x - μ)^2 * P(x))
Variance (σ^2) = ((0 - 1.199)^2 * 0.167) + ((1 - 1.199)^2 * 0.500) + ((2 - 1.199)^2 * 0.300) + ((3 - 1.199)^2 * 0.033)
Variance (σ^2) = (1.199^2 * 0.167) + (0.199^2 * 0.500) + (0.801^2 * 0.300) + (1.801^2 * 0.033)
Variance (σ^2) = 0.264 + 0.0198 + 0.192 + 0.014
Variance (σ^2) = 0.49
Step 3: Finally, take the square root of the variance to find the standard deviation:
Standard Deviation (σ) = √(Variance)
Standard Deviation (σ) = √(0.49)
Standard Deviation (σ) = 0.70
So the standard deviation of x is 0.70.
To construct the probability distribution table for the random variable x, which represents the number of rotten apples chosen, you can follow these steps:
Step 1: Determine the possible values for x. In this case, x can take on values from 0 to 3 since there are only four rotten apples.
Step 2: Calculate the probability of each value of x.
For x = 0: The probability of choosing zero rotten apples is determined by the number of ways to choose 3 healthy apples out of the 5 remaining good apples divided by the total number of ways to choose 3 apples out of the 9 total apples. The calculation is (C(5,3) / C(9,3)) = (10/84) = 0.119.
For x = 1: The probability of choosing one rotten apple is the number of ways to pick 1 rotten apple out of 4, multiplied by the number of ways to pick 2 healthy apples out of the remaining 5 good apples, divided by the total number of ways to choose 3 apples out of the 9 total apples. The calculation is (C(4,1) * C(5,2)) / C(9,3) = (4 * 10) / 84 = 0.476.
For x = 2: The probability of choosing two rotten apples is the number of ways to pick 2 rotten apples out of 4, multiplied by the number of ways to pick 1 healthy apple out of the remaining 5 good apples, divided by the total number of ways to choose 3 apples out of the 9 total apples. The calculation is (C(4,2) * C(5,1)) / C(9,3) = (6 * 5) / 84 = 0.357.
For x = 3: The probability of choosing all three rotten apples is the number of ways to pick 3 rotten apples out of 4, divided by the total number of ways to choose 3 apples out of the 9 total apples. The calculation is (C(4,3) / C(9,3)) = (4/84) = 0.048.
Step 3: Summarize the results in a table, with the possible values of x and their respective probabilities, as shown in the question:
X P(x)
0 0.119
1 0.476
2 0.357
3 0.048
To find the mean of the random variable x, you can multiply each value of x by its respective probability and sum them up:
Mean = (0 * 0.119) + (1 * 0.476) + (2 * 0.357) + (3 * 0.048) = 0.952
To find the standard deviation, you need to calculate the variance first. The variance is the sum of each squared difference between x and the mean, multiplied by its respective probability:
Variance = [(0-0.952)^2 * 0.119] + [(1-0.952)^2 * 0.476] + [(2-0.952)^2 * 0.357] + [(3-0.952)^2 * 0.048] = 0.943
Finally, the standard deviation can be found by taking the square root of the variance:
Standard Deviation = √(0.943) = 0.971