91Ó°ÊÓ

The combination formula is a crucial concept in probability and statistics. It helps in determining the number of ways to pick a group of items from a larger set, where the order does not matter. This is perfect for cases like selecting samples or groups without concern for sequence.

For our exercise, the formula is denoted as \( \binom{n}{k} \), which reads: "n choose k." Here, \( n \) is the total number of items to choose from, and \( k \) is the number of items to select. The formula is:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

The exclamation mark "!" denotes a factorial, which means multiplying a number by all positive integers less than it. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

In this problem, to find out how many ways we can pick 3 totes out of 15, we use it as: \( \binom{15}{3} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = 455 \). So, there are 455 different combinations of totes possible.

Probability calculation is used to find the likelihood that an event will happen. Here, we are interested in finding the probability of selecting at least one nonconforming tote from the sample of totes.

To determine this probability, we first calculate the number of favorable outcomes, which is selecting at least one nonconforming tote. This is done by subtracting the cases where all selected totes conform from the total selections.

In mathematical terms, the probability \( P \) can be formulated as:

• \( P(\text{at least one nonconforming}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} \)

In the exercise, the calculated favorable outcomes were 169, and the total outcomes were 455. So:

• \( P = \frac{169}{455} = \frac{13}{35} \)

This fraction is the simplified form, representing the probability that at least one of the selected totes is nonconforming.

Sample selection is the process of selecting a subset from a larger set. It's fundamental in statistics, especially when dealing with a large population, to estimate probabilities or test hypotheses via a smaller sample.

In the context of this exercise, the entire shipment consists of 15 totes, from which 3 are selected randomly (without replacement) for inspection.

Key considerations in sample selection include:

• Size of the sample: Here, 3 totes are chosen.
• Randomness: The selection must be random to avoid bias.
• Replacement: Since the selection is without replacement, once a tote is selected, it cannot be chosen again for the same sample.

The randomness and lack of replacement directly affect the sample's reliability in representing the whole population.

Nonconforming items do not meet the established quality or criteria. In quality control processes, identifying nonconforming items is essential to ensuring product standards.

For this problem, nonconforming items are defined as those that do not meet the purity requirements. Among the 15 totes received, 2 have been identified as nonconforming.

Why worry about nonconforming items?

• Ensures quality assurance by detecting faults early.
• Helps in maintaining consistency across the product line.
• Reduces waste by identifying potential issues before they reach the customer.

Understanding the makeup of conforming and nonconforming totes is vital to calculating the probability and ensuring an unbiased inspection process. The probability problem presented here is about quantifying the likelihood of encountering these nonconforming totes during random selection.