A shipment of 9 printers contains 4 that are defective. Find the probability that a sample of size 4, drawn from the 9, will not contain a defective printer .
the probability is ?
there are 9C4 ways of selecting four printers
there are 5C4 ways of selecting four good printers
the probability is ... 5C4 / 9C4
To calculate the probability that a sample of size 4, drawn from the 9 printers, will not contain a defective printer, we can use the concept of combinations.
The total number of ways to choose 4 printers from 9 is given by the combination formula:
C(n, r) = n! / (r! * (n-r)!)
Where n is the total number of printers (9 in this case) and r is the size of the sample (4 in this case).
The number of ways to choose 4 printers from the 5 non-defective printers is given by C(5, 4):
C(5, 4) = 5! / (4! * (5-4)!) = 5
Therefore, the probability of not choosing a defective printer is:
P(not defective) = C(5, 4) / C(9, 4)
= 5 / C(9, 4)
= 5 / (9! / (4! * (9-4)!))
= 5 / (9! / (4! * 5!))
= 5 / (9 * 8 * 7 * 6 / (4 * 3 * 2 * 1 * 5))
= 5 / (126/120)
= 5 / (7/6)
= 5 * 6 / 7
= 30 / 7
Therefore, the probability that a sample of size 4, drawn from the 9 printers, will not contain a defective printer is 30/7, which is approximately 4.29.
To find the probability that a sample of size 4, drawn from the 9 printers, will not contain a defective printer, we need to first calculate the total number of possible samples of size 4. Then, we will calculate the number of samples that do not contain a defective printer.
The total number of possible samples of size 4 from a set of 9 printers can be calculated using the combination formula, which is defined as:
C(n, r) = n! / (r!(n - r)!)
where n is the total number of objects, r is the number of objects in each sample, and ! denotes the factorial function.
In this case, n = 9 and r = 4. Let's calculate it:
C(9, 4) = 9! / (4!(9 - 4)!)
= 9! / (4!5!)
= (9 * 8 * 7 * 6 * 5!) / (4! * 5!)
= (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1)
= 126
So, there are 126 possible samples of size 4 that can be drawn from the 9 printers.
Now, let's calculate the number of samples that do not contain a defective printer. Since there are 4 defective printers out of 9, the number of non-defective printers is 9 - 4 = 5. So, we need to calculate the combinations of 4 non-defective printers from the set of 5 non-defective printers:
C(5, 4) = 5! / (4!(5 - 4)!)
= 5! / (4! * 1!)
= 5
Therefore, there are 5 possible samples of size 4 that do not contain a defective printer.
Finally, to find the probability, we divide the number of samples that do not contain a defective printer by the total number of possible samples:
Probability = Number of favorable outcomes / Number of total outcomes
= 5 / 126
= 0.0397 (approx.)
So, the probability that a sample of size 4, drawn from the 9 printers, will not contain a defective printer is approximately 0.0397.