A batch contains 36 bacteria cells and 12 of the cells are...

A batch contains 36 bacteria cells and 12 of the cells are not capable of cellular replication. Suppose that you examine three bacteria cells selected at random without replacement. (a) What is the probability mass function of the number of cells in the sample that can replicate? (b) What are the mean and variance of the number of cells in the sample that can replicate? (c) What is the probability that at least one of the selected cells cannot replicate?

A batch contains 36 bacteria cells and 12 of the cells are not capable of cellular replication. Suppose that you examine three bacteria cells selected at random without replacement.
(a) What is the probability mass function of the number of cells in the sample that can replicate?
(b) What are the mean and variance of the number of cells in the sample that can replicate?
(c) What is the probability that at least one of the selected cells cannot replicate?

Key Concepts

Hypergeometric Distribution

This distribution is used when sampling without replacement from a finite population. It describes the probability of drawing a specified number of successes (in this case, replicable cells) from a population that contains a known number of successes and failures. The key parameters of the hypergeometric distribution are the population size, number of successes in the population, and the sample size.

Probability Mass Function (PMF)

The PMF is the function that provides the probability of a discrete random variable taking on a particular value. For the hypergeometric distribution, the PMF calculates the probability of drawing exactly k successes in the sample and is defined using combinatorial expressions that account for drawing successes and failures without replacement.

Mean and Variance of the Hypergeometric Distribution

The mean and variance formulas for the hypergeometric distribution quantify the expected number of successes and the spread around that expectation. The mean is determined by both the proportion of successes in the population and the sample size, while the variance accounts for the finite population correction factor, reflecting the reduced variability when sampling without replacement.

Complement Rule

The complement rule in probability is used to find the probability of the complement of an event by subtracting the probability of the event from one. In the context of the problem, it is used to calculate the probability that at least one cell in the selected sample is non-replicative by subtracting the probability that all selected cells are replicative from one.

Please give Ace some feedback