Suppose you work for the quality control department at Hewlett Packard. You know that ink jet cartridges are sometimes defective. The current defective rate is 8 cartridges per package of 100 (i.e. 0.08) What is the probability of sampling 12 cartridges and finding:
At least 4 are defective:
Exactly 1 is defective:
2 or more, but less than 5 is defective:
These are the main ones I'm unsure on, any help would be greatly appreciated.
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I'm sorry but I just don't understand.
each term is the probability of that specific outcome
n^x d^y means x were not defective and y were defective
... in this problem , x + y = 12
at least 4 defective means the sum of the terms where 4 or more were defective
answers for these three were covered in your 1st posting
To find the probabilities of different scenarios, we can use the binomial probability formula. The binomial distribution is applicable since this is a situation with a fixed number of trials (sampling 12 cartridges) and each trial has two possible outcomes (defective or not defective).
The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability that exactly k out of n trials are successful,
n is the total number of trials (12 in this case),
k is the number of successful trials we are interested in,
p is the probability of success in a single trial (defect rate - 0.08 in this case),
C(n, k) represents the binomial coefficient or "n choose k" which is the number of ways k successes can be chosen from n trials.
Let's calculate the probabilities for the given scenarios:
1. At least 4 cartridges are defective:
To find this probability, we need to calculate the sum of probabilities for having 4, 5, 6... up to 12 defective cartridges. We can use the complement rule and subtract from 1 the probability of having 0, 1, 2, or 3 defective cartridges.
P(at least 4 defective) = 1 - P(0 defective) - P(1 defective) - P(2 defective) - P(3 defective)
2. Exactly 1 cartridge is defective:
P(exactly 1 defective) = P(X = 1)
3. 2 or more, but less than 5 cartridges are defective:
To find this probability, we need to calculate the sum of probabilities for having 2, 3, or 4 defective cartridges.
P(2 ≤ X < 5) = P(X = 2) + P(X = 3) + P(X = 4)
By substituting the given values into the binomial probability formula, we can calculate the probabilities for each scenario.