You work in quality control at a rod manufacturing plant. You want to analyze the quality of rod batches using the following probability rules and descriptive statistics.

## Core Formulas

### Union Rule

$$
P(A \cup B) = P(A) + P(B) - P(A \cap B)
$$

### Conditional Probability

$$
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
$$

### Sample Statistics

- Sample Mean:  
  $$
  \bar{x} = \frac{\sum x_i}{n}
  $$

- Sample Variance:  
  $$
  s^2 = \frac{\sum (x_i - \bar{x})^2}{n}
  $$

- Sample Standard Deviation:  
  $$
  s = \sqrt{s^2}
  $$

## Given Data

- Total rods: 100; 
- Defective rods: 20;
- Rods longer than 50 cm: 30;  
- Rods that are both defective and long: 10;
- Population mean rod length: 50 cm;
- Population standard deviation of rod lengths: 0.5 cm;
- Sample size for analysis: 10 rods.


import unittest
import numpy as np
import user_code  # ÐºÐ¾Ð´ ÑÑÑÐ´ÐµÐ½ÑÐ°

def _dynamic(tc, ok, ok_msg, fail_msg):
    tc._testMethodName = ok_msg if ok else fail_msg
    (tc.assertTrue if ok else tc.fail)(tc._testMethodName)

ATOL = 1e-10
RTOL = 1e-9

class TestQualityControlProb(unittest.TestCase):

    def test_variables_exist_and_types(self):
        reasons = []

        # Scalars & counts
        total_rods = getattr(user_code, "total_rods", None)
        defective = getattr(user_code, "defective", None)
        long_ = getattr(user_code, "long", None)
        defective_and_long = getattr(user_code, "defective_and_long", None)

        # Probabilities
        P_D_or_L = getattr(user_code, "P_D_or_L", None)
        P_D_given_L = getattr(user_code, "P_D_given_L", None)

        # Sampling params & outputs
        population_mean = getattr(user_code, "population_mean", None)
        population_std = getattr(user_code, "population_std", None)
        sample_size = getattr(user_code, "sample_size", None)
        sample_lengths = getattr(user_code, "sample_lengths", None)
        sample_mean = getattr(user_code, "sample_mean", None)
        sample_variance = getattr(user_code, "sample_variance", None)
        sample_std = getattr(user_code, "sample_std", None)

        # Basic presence & type checks
        if not isinstance(total_rods, (int, np.integer)):
            reasons.append("total_rods must be int")
        if not isinstance(defective, (int, np.integer)):
            reasons.append("defective must be int")
        if not isinstance(long_, (int, np.integer)):
            reasons.append("long must be int")
        if not isinstance(defective_and_long, (int, np.integer)):
            reasons.append("defective_and_long must be int")

        if not isinstance(P_D_or_L, (float, np.floating)):
            reasons.append("P_D_or_L must be float")
        if not isinstance(P_D_given_L, (float, np.floating)):
            reasons.append("P_D_given_L must be float")

        if not isinstance(population_mean, (int, float, np.integer, np.floating)):
            reasons.append("population_mean must be numeric")
        if not isinstance(population_std, (int, float, np.integer, np.floating)):
            reasons.append("population_std must be numeric")
        if not isinstance(sample_size, (int, np.integer)):
            reasons.append("sample_size must be int")

        if not (isinstance(sample_lengths, np.ndarray) and sample_lengths.shape == (sample_size,)):
            reasons.append(f"sample_lengths must be ndarray of shape ({sample_size},)")
        if not isinstance(sample_mean, (float, np.floating)):
            reasons.append("sample_mean must be float")
        if not isinstance(sample_variance, (float, np.floating)):
            reasons.append("sample_variance must be float")
        if not isinstance(sample_std, (float, np.floating)):
            reasons.append("sample_std must be float")

        _dynamic(self,
                 len(reasons) == 0,
                 "PASS: variables exist with correct types/shapes",
                 "FAIL: " + " | ".join(reasons) if reasons else "FAIL")

    def test_probabilities_correct(self):
        reasons = []

        total_rods = user_code.total_rods
        defective = user_code.defective
        long_ = user_code.long
        defective_and_long = user_code.defective_and_long

        expected_union = (defective/total_rods) + (long_/total_rods) - (defective_and_long/total_rods)
        expected_cond = (defective_and_long/total_rods) / (long_/total_rods)

        if not np.isclose(user_code.P_D_or_L, expected_union, atol=ATOL, rtol=RTOL):
            reasons.append(f"P(D âª L) incorrect: expected {expected_union}, got {user_code.P_D_or_L}")
        if not np.isclose(user_code.P_D_given_L, expected_cond, atol=ATOL, rtol=RTOL):
            reasons.append(f"P(D|L) incorrect: expected {expected_cond}, got {user_code.P_D_given_L}")

        _dynamic(self,
                 len(reasons) == 0,
                 "PASS: probabilities (union & conditional) are correct",
                 "FAIL: " + " | ".join(reasons) if reasons else "FAIL")

    def test_sampling_reproducible_and_stats(self):
        reasons = []

        # Recompute expected sample deterministically with the same seed & params
        population_mean = user_code.population_mean
        population_std = user_code.population_std
        sample_size = user_code.sample_size

        np.random.seed(0)
        expected_sample = np.random.normal(loc=population_mean, scale=population_std, size=sample_size)
        expected_mean = np.mean(expected_sample)
        expected_var = np.var(expected_sample, ddof=1)
        expected_std = np.std(expected_sample, ddof=1)

        # Compare arrays & stats
        if not (isinstance(user_code.sample_lengths, np.ndarray) and user_code.sample_lengths.shape == expected_sample.shape):
            reasons.append("sample_lengths shape mismatch")
        else:
            if not np.allclose(user_code.sample_lengths, expected_sample, atol=1e-12, rtol=1e-12):
                reasons.append("sample_lengths not reproducible with seed=0 and given parameters")

        if not np.isclose(user_code.sample_mean, expected_mean, atol=ATOL, rtol=RTOL):
            reasons.append(f"sample_mean incorrect: expected {expected_mean}, got {user_code.sample_mean}")
        if not np.isclose(user_code.sample_variance, expected_var, atol=ATOL, rtol=RTOL):
            reasons.append(f"sample_variance incorrect (use ddof=1): expected {expected_var}, got {user_code.sample_variance}")
        if not np.isclose(user_code.sample_std, expected_std, atol=ATOL, rtol=RTOL):
            reasons.append(f"sample_std incorrect (use ddof=1): expected {expected_std}, got {user_code.sample_std}")

        _dynamic(self,
                 len(reasons) == 0,
                 "PASS: sampling is reproducible & stats (mean/var/std) correct",
                 "FAIL: " + " | ".join(reasons) if reasons else "FAIL")


test_code.py

Master the mathematical foundations essential for data science. Explore core concepts in functions, calculus, linear algebra, probability, and dimensionality reduction. Build both theoretical understanding and practical coding experience to strengthen your ability to analyze data, model complex systems, and apply advanced techniques in machine learning.

Explore the foundation of mathematical functions. Learn different types of algebraic and transcendental functions, their properties, and how to implement them in Python to solve real-world problems.

Master the concepts of sets and series, from basic operations to practical applications. Gain hands-on experience implementing set operations and working with arithmetic and geometric series in Python.

Develop a solid understanding of limits, derivatives, integrals, and partial derivatives. Connect theory to practice by implementing these concepts in Python and applying them to optimization through gradient descent.

Build strong knowledge of vectors, matrices, and transformations. Learn decomposition methods and eigenvalue analysis, while reinforcing concepts with Python coding challenges and practical data science applications.

Dive into probability theory and statistics. Study conditional probability, Bayes' theorem, and statistical measures. Implement key concepts in Python, simulate distributions, and strengthen your skills through challenges and quizzes.

Challenge: Quality Control Probability Analysis

Core Formulas

Union Rule

Conditional Probability

Sample Statistics

Given Data

Oplossing

Awesome!