Math Problem Statement

Determine the probability that at most 2 products out of 4 sampled units belong to the 'Good' class and estimate the mean and standard deviation of the number of 'Good' products in a sample of 4 units.

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:

  1. Binomial Distribution Formula: The binomial probability mass function is given by: P(X=k)=(nk)pk(1p)nkP(X = k) = \binom{n}{k} p^k (1-p)^{n-k} where nn is the number of trials (4), pp is the probability of success (0.70), and kk is the number of successful outcomes.

  2. Cumulative Probability: For task 1, you'll need to calculate: P(X2)=P(X=0)+P(X=1)+P(X=2)P(X \leq 2) = P(X=0) + P(X=1) + P(X=2)

  3. Mean and Standard Deviation:

    • The mean (μ) of a binomial distribution is given by: μ=n×p\mu = n \times p
    • The standard deviation (σ) is: σ=n×p×(1p)\sigma = \sqrt{n \times p \times (1 - p)}

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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