Q: What are the conditions for hypergeometric distribution?

Conditions required: 1) Population size N is finite and known. 2) There are exactly two outcomes (success/failure). 3) Sampling is done without replacement. 4) Sample size n is fixed. 5) Each item has equal probability of being selected. Additionally, parameters must satisfy: K ≤ N (successes ≤ population), n ≤ N (sample ≤ population), and k ≤ min(n, K) (sample successes ≤ both sample size and available successes).

Q: How is hypergeometric distribution used in quality control?

In quality control, hypergeometric distribution calculates the probability of finding defective items in a sample from a finite batch. Example: A shipment has 100 items with 10 defective (N=100, K=10). You inspect 20 items (n=20). What's the probability of finding exactly 3 defective items (k=3)? Use hypergeometric formula to calculate P(X=3). This helps determine acceptance/rejection criteria for lot sampling and quality inspection procedures.

Question 1

What is hypergeometric distribution?

Accepted Answer

Hypergeometric distribution is a discrete probability distribution that describes the probability of k successes in n draws from a finite population of size N containing K success states, without replacement. Unlike the binomial distribution which assumes replacement (or infinite population), hypergeometric distribution accounts for changing probabilities as items are drawn. It's used when sampling from small, finite populations.

Question 2

What is the hypergeometric probability formula?

Accepted Answer

The hypergeometric probability formula is: P(X = k) = [C(K,k) × C(N-K, n-k)] / C(N,n), where N is population size, K is number of successes in population, n is sample size, k is number of successes in sample, and C(n,k) is the combination function 'n choose k'. This formula calculates the probability of getting exactly k successes when drawing n items from a population of N items containing K successes.

Question 3

When should you use hypergeometric distribution instead of binomial?

Accepted Answer

Use hypergeometric distribution when: 1) Sampling without replacement from a finite population. 2) Population size is relatively small compared to sample size (usually n > 5% of N). 3) Probability of success changes with each draw. Use binomial distribution when: 1) Sampling with replacement or population is very large. 2) Probability remains constant. 3) Trials are independent. Example: Drawing cards without replacement = hypergeometric; coin flips = binomial.

Question 4

How do you calculate hypergeometric probability?

Accepted Answer

Step 1: Identify N (population), K (successes in population), n (sample size), k (desired successes). Step 2: Calculate C(K,k) - ways to choose k successes from K. Step 3: Calculate C(N-K, n-k) - ways to choose remaining items from failures. Step 4: Calculate C(N,n) - total ways to choose sample. Step 5: Probability = [C(K,k) × C(N-K,n-k)] / C(N,n). The combination C(n,k) = n! / (k!(n-k)!).

Question 5

What is the mean of hypergeometric distribution?

Accepted Answer

The mean (expected value) of a hypergeometric distribution is μ = n × K / N, where n is the sample size, K is the number of successes in the population, and N is the population size. This represents the expected number of successes in your sample. For example, if you draw 5 cards from a deck (N=52) of which 13 are hearts (K=13), you expect μ = 5 × 13/52 = 1.25 hearts on average.

Question 6

How do you calculate variance for hypergeometric distribution?

Accepted Answer

The variance formula for hypergeometric distribution is: σ² = n × K × (N-K) × (N-n) / [N² × (N-1)], where n is sample size, K is successes in population, N is population size. The term (N-n)/(N-1) is the finite population correction factor that accounts for sampling without replacement. Standard deviation is σ = √(variance). Variance measures the spread of the distribution around the mean.

Question 7

What is a real-world example of hypergeometric distribution?

Accepted Answer

Classic example: Drawing cards from a deck without replacement. If you want the probability of getting exactly 2 hearts in a 5-card hand: N=52 total cards, K=13 hearts, n=5 cards drawn, k=2 hearts desired. P(X=2) ≈ 0.2743 or 27.43%. Other examples include: quality control (selecting defective items from a batch), lottery drawings, genetics (allele sampling), and survey sampling from small populations.

Question 8

What is the difference between hypergeometric and binomial distribution?

Accepted Answer

Key differences: Hypergeometric uses sampling WITHOUT replacement (finite population), probabilities change with each draw, used for small populations. Binomial uses sampling WITH replacement (or infinite population), probabilities remain constant, trials are independent. If population is large relative to sample (n < 5% of N), hypergeometric approximates binomial. Hypergeometric is more accurate for small, finite populations.

Question 9

What are the conditions for hypergeometric distribution?

Accepted Answer

Conditions required: 1) Population size N is finite and known. 2) There are exactly two outcomes (success/failure). 3) Sampling is done without replacement. 4) Sample size n is fixed. 5) Each item has equal probability of being selected. Additionally, parameters must satisfy: K ≤ N (successes ≤ population), n ≤ N (sample ≤ population), and k ≤ min(n, K) (sample successes ≤ both sample size and available successes).

Question 10

How is hypergeometric distribution used in quality control?

Accepted Answer

In quality control, hypergeometric distribution calculates the probability of finding defective items in a sample from a finite batch. Example: A shipment has 100 items with 10 defective (N=100, K=10). You inspect 20 items (n=20). What's the probability of finding exactly 3 defective items (k=3)? Use hypergeometric formula to calculate P(X=3). This helps determine acceptance/rejection criteria for lot sampling and quality inspection procedures.

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