QuestionA factory has 22 compressors with 5 defective. Find the probabilities for a sample of 5: all defective and none defective.
Studdy Solution
STEP 1
Assumptions1. The total number of compressors is22.
. The number of defective compressors is5.
3. The number of compressors selected for the sample is5.
4. Each compressor is equally likely to be selected.
STEP 2
First, we need to find the total number of ways to select5 compressors from22. This is a combination problem, which can be solved using the combination formulawhere n is the total number of items, k is the number of items to select, and "!" denotes factorial.
STEP 3
Now, plug in the given values for n and k to calculate the total number of ways to select5 compressors from22.
STEP 4
Calculate the total number of ways to select compressors from22.
STEP 5
Now, we need to find the number of ways to select5 defective compressors from5. This is also a combination problem.
STEP 6
Calculate the number of ways to select5 defective compressors from5.
STEP 7
The probability that all in the sample are defective is the ratio of the number of ways to select5 defective compressors from5 to the total number of ways to select5 compressors from22.
STEP 8
Plug in the values for the number of ways to select5 defective compressors from5 and the total number of ways to select5 compressors from22 to calculate the probability.
STEP 9
Now, we need to find the number of ways to select5 non-defective compressors from17 (since there are22 total compressors and5 are defective, leaving17 non-defective). This is also a combination problem.
STEP 10
Calculate the number of ways to select5 non-defective compressors from17.
STEP 11
The probability that none in the sample are defective is the ratio of the number of ways to select5 non-defective compressors from17 to the total number of ways to select5 compressors from22.
STEP 12
Plug in the values for the number of ways to select5 non-defective compressors from17 and the total number of ways to select5 compressors from22 to calculate the probability.
STEP 13
implify the fraction to get the reduced form.
The probability that all in the sample are defective is .
The probability that none in the sample are defective is .
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