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In probability theory, a Probability Mass Function (PMF) is a function that provides the probability of each possible value occurring in a random variable's distribution. For discrete variables such as those in a hypergeometric distribution, the PMF will give us the likelihood of a certain number of items (e.g., granite specimens) being selected in a sample.

For the hypergeometric distribution, the PMF can be calculated using:

• \[ P(X=k) = \frac{{\binom{m}{k} \binom{N-m}{n-k}}}{\binom{N}{n}} \]
• Here, \( N \) is the total number of specimens (20),
• \( m \) is the number of granite specimens (10),
• and \( n \) is the number of specimens selected (15).

To compute this, you would substitute \( k \) (the number of granite specimens selected) for values like 0, 1, 2, up to \( m \). This formula helps us to precisely calculate the likelihood of drawing a specific number of granite specimens.

The mean, or expected value, of a hypergeometric distribution represents the average number of successes (granite specimens, in this case) expected when a sample is drawn without replacement from a finite population.

For a hypergeometric distribution, the mean \( \mu \) can be calculated with the formula:

• \[ \mu = n \left( \frac{m}{N} \right) \]
• With our existing values, we calculate:
• \( n = 15 \) (specimens selected),
• \( m = 10 \) (granite specimens),
• \( N = 20 \) (total specimens).

Therefore, \( \mu = 15 \times \left( \frac{10}{20} \right) = 7.5 \). This tells us that, on average, we expect to select 7.5 granite specimens when choosing 15 specimens from a total of 20.

Variance and standard deviation measure the spread or dispersion of a set of values. In the context of hypergeometric distribution, these measure how much the number of granite specimens can differ from the mean.

The variance \( \sigma^2 \) for a hypergeometric distribution is calculated using:

• \[ \sigma^2 = n \left( \frac{m}{N} \right) \left( \frac{N-m}{N} \right) \left( \frac{N-n}{N-1} \right) \]
• Substitute:
  • \( n = 15 \)
  • \( m = 10 \)
  • \( N = 20 \)
• This gives \( \sigma^2 = 15 \times \frac{10}{20} \times \frac{10}{20} \times \frac{5}{19} \).

To find the standard deviation \( \sigma \), take the square root of the variance. This helps provide a sense of how varied or spread out the sample size can be centered around the mean.

Sampling without replacement is a method used in probability and statistics where each selected item is not put back into the population before the next draw. This method impacts the probabilities and is essential in understanding hypergeometric distributions.

When sampling without replacement, the probability of each item being selected is not constant since the total number of items in the population decreases with each draw. This behavior contrasts with sampling with replacement, where probabilities remain unchanged. In this specific problem, no specimen is returned once selected, which means the combinations of granite and basalt specimens selected can vary with each attempt.

This technique is crucial in many real-world scenarios. It accurately reflects many situations where resources can be used up as they are selected, such as dealing out a fixed number of cards or drawing specimens for analysis without replacing them back, yielding a more realistic model of randomness in finite populations.