Math Problem Statement
Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 25 failed keyboard, 7 or which have electrical defects and 18 of which have mechanical defects. If a sample of 7 keyboards is randomly selected, what is the probability that at least 6 of these will have a mechanical defect? Round your answer to four decimal places
Solution
To solve this problem, we can model the situation using the hypergeometric distribution because we are sampling without replacement from a finite population.
Hypergeometric Distribution Formula:
The probability of selecting items of a certain type in a sample of size from a population of items, where are of the desired type, is given by:
Where:
- = Total population size (25 keyboards)
- = Total number of mechanical defect keyboards (18 keyboards)
- = Sample size (7 keyboards)
- = Number of mechanical defect keyboards in the sample
- We want to find the probability that at least 6 of the 7 keyboards have mechanical defects.
Probability Calculation:
We need to calculate , which is the sum of the probabilities for and :
Step 1: Calculate
Step 2: Calculate
Now, let's compute these probabilities.The probability that at least 6 out of the 7 selected keyboards will have a mechanical defect is approximately (rounded to four decimal places).
Would you like more details or have any questions about this solution?
Here are 5 related questions to further explore the topic:
- What is the probability of selecting exactly 5 keyboards with mechanical defects?
- How would the probability change if the sample size were 5 keyboards instead of 7?
- Can we calculate the expected number of mechanical defects in a random sample of 7 keyboards?
- What if 10 keyboards had electrical defects instead of 7? How would that impact the probability?
- How does the hypergeometric distribution differ from the binomial distribution in this context?
Tip: The hypergeometric distribution is useful when sampling without replacement, while the binomial distribution is better for sampling with replacement.
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Hypergeometric Distribution
Formulas
P(X = k) = (C(K, k) * C(N - K, n - k)) / C(N, n)
P(X >= 6) = P(X = 6) + P(X = 7)
Theorems
Hypergeometric Distribution Theorem
Suitable Grade Level
Undergraduate level (or Advanced High School)
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