Final answer:
To find the probability that two of the ten randomly selected modems are defective, we use the hypergeometric distribution formula, combining the ways to choose defective and non-defective modems and dividing by the total ways to select any ten modems.
Step-by-step explanation:
The problem presented involves calculating the probability that exactly two out of ten randomly selected modems from a shipment of sixteen, of which three are defective, will be defective. This is a classic example of a hypergeometric distribution problem because the samples are drawn without replacement from a finite population that contains a known number of defects.
First, we determine the number of ways to choose two defective modems out of the three available. This is given by the combination formula C(3,2). Next, we find the number of ways to select the remaining eight modems from the thirteen non-defective ones, which is C(13,8). Finally, we calculate the total number of ways to choose ten modems from the sixteen, which is C(16,10).
The probability is then found by dividing the product of the number of ways to select the defective modems and the number of ways to select the non-defective ones by the total number of ways to select any ten modems.
Therefore, the probability is computed as:
P(2 defective) = [C(3,2) * C(13,8)] / C(16,10)