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The Probability Mass Function (PMF) in statistics refers to a function that provides the probability of each possible value for a discrete random variable. In the context of the hypergeometric distribution, the PMF is particularly useful in determining the probability of observing a certain number of successful outcomes in a sample drawn without replacement. When dealing with a limited population, like in our bacteria cell example, each draw affects the next one, making the hypergeometric model ideal.

To find the PMF for the number of replicating cells out of a sample of three examined, we use the hypergeometric distribution formula:

• Let - \( N \) represent the total number of bacteria cells (36). - \( K \) be the number of replicating cells (24). - \( n \) interchangeably signify the sample size (3). - \( k \) is the specific number of replicating cells in our sample.

The PMF for a hypergeometric distribution is calculated as:\[ P(X = k) = \frac{\binom{K}{k} \cdot \binom{N-K}{n-k}}{\binom{N}{n}} \]
Applying our values, you calculate the probabilities for every scenario from 0 to 3 replicating cells. This step is crucial, as it outlines all potential occurrences and their respective probabilities, forming the basis for subsequent calculations of mean and variance.

In probability and statistics, calculating the mean and variance provides insight into the expected outcomes and variability respectively. For the hypergeometric distribution, these calculations tailor to scenarios like our bacteria cell selection.

Mean of the Hypergeometric Distribution

The mean, also referred to as the expected value, of the hypergeometric distribution provides the average number of successful observations (replicating bacteria in this case) over many samples. The formula is:\[ \mu = n\frac{K}{N} \]Where

• \( n \) is the sample size (3).
• \( K \) denotes the number of successful elements (24).
• \( N \) is the total number of elements (36).

By inserting these values, we find the mean \( \mu \) to be 2, indicating on average, you expect to find 2 replicating cells in a random selection of three.

Variance of the Hypergeometric Distribution

Variance measures how much the outcomes in a probability distribution spread around the mean. For this distribution, the variance \( \sigma^2 \) is calculated as:\[ \sigma^2 = n \frac{K}{N} \frac{N-K}{N} \frac{N-n}{N-1} \]Using our values, the variance comes out to approximately 0.6943. This reflects the deviation you might expect from the mean number of successful outcomes.

Complementary probability is a logical method used to find the probability of an event by subtracting the probability of its complement from one. In our exercise, we're interested in finding the likelihood that at least one of the selected bacteria cells cannot replicate.

The complement of selecting at least one non-replicating cell is that all cells selected can replicate. First, compute the probability for this case, which is when all three (\( n=3 \)) chosen are replicating. Known from previous calculations, this is \( P(X = 3) = 0.2837 \).

Then, using complementary probability, the probability of having at least one non-replicating cell is:

• \[ P(\text{at least 1 non-replicating}) = 1 - P(X = 3) = 1 - 0.2837 = 0.7163 \]

This approach is often simpler when determining probabilities involving phrases like "at least" or "at most," making complex calculations more manageable.