Math Problem Statement
Solution
The image contains a problem related to statistics and probability, specifically focused on binomial distribution. Here's a breakdown of the problem:
Problem Description:
A product's lifecycle is categorized into four classes: Failed on Arrival (FOA), Good, Low, and High. The corresponding probabilities for each class are provided in a table.
- FOA: 0.05
- Good: 0.70
- Low: 0.15
- High: 0.10
Task 1: Determine the probability that at most 2 products out of 4 sampled units belong to the Good class. This involves calculating the cumulative probability of getting 0, 1, or 2 "Good" products in a binomial distribution where the probability of success (being in the Good class) is 0.70.
Task 2: Estimate the mean and standard deviation of the number of "Good" products in a sample of 4 units.
Solution Approach:
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Binomial Distribution Formula: The binomial probability mass function is given by: where is the number of trials (4), is the probability of success (0.70), and is the number of successful outcomes.
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Cumulative Probability: For task 1, you'll need to calculate:
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Mean and Standard Deviation:
- The mean (μ) of a binomial distribution is given by:
- The standard deviation (σ) is:
Would you like me to calculate the exact values for these probabilities and parameters?
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Math Problem Analysis
Mathematical Concepts
Binomial Distribution
Probability
Statistics
Formulas
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
P(X <= 2) = P(X=0) + P(X=1) + P(X=2)
Mean (μ) = n * p
Standard deviation (σ) = sqrt(n * p * (1 - p))
Theorems
Binomial Theorem
Suitable Grade Level
Grades 10-12
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