Problem 41 - Applied Statistics and Probability for Engineers

Question

A batch of 140 semiconductor chips is inspected by choosing a sample of 5 chips. Assume 10 of the chips do not conform to customer requirements. (a) How many different samples are possible? (b) How many samples of five contain exactly one nonconforming chip? (c) How many samples of five contain at least one nonconforming chip?

Step-by-Step Solution

Answer

(a) 4,479,848 samples (b) 28,563,360 samples (c) 1,689,344 samples

Step 1: Total Number of Samples

To determine the total number of different samples possible, we need to find the number of ways to choose 5 chips out of 140. This can be calculated using the combination formula, \( \binom{n}{k} \), which represents the number of combinations of \( n \) items taken \( k \) at a time. Thus, the total number of samples is \( \binom{140}{5} \). Calculate \( \binom{140}{5} \):\[ \binom{140}{5} = \frac{140 \times 139 \times 138 \times 137 \times 136}{5 \times 4 \times 3 \times 2 \times 1} = 4,479,848 \].

Step 2: Samples Containing Exactly One Nonconforming Chip

To find the number of samples containing exactly one nonconforming chip, we first choose 1 nonconforming chip from the 10 available, and then choose 4 conforming chips from the 130 available. The number of ways to choose 1 nonconforming chip is \( \binom{10}{1} = 10 \). The number of ways to choose 4 conforming chips is \( \binom{130}{4} \). Calculate \( \binom{130}{4} \):\[ \binom{130}{4} = \frac{130 \times 129 \times 128 \times 127}{4 \times 3 \times 2 \times 1} = 2,856,336 \].Thus, the total number of samples with exactly one nonconforming chip is:\[ 10 \times 2,856,336 = 28,563,360 \].

Step 3: Samples Containing At Least One Nonconforming Chip

To find the number of samples with at least one nonconforming chip, we can use complementary counting. First, find the number of samples that contain only conforming chips (none of the 10 nonconforming). This is equivalent to choosing 5 chips from the 130 conforming chips. Calculate \( \binom{130}{5} \):\[ \binom{130}{5} = \frac{130 \times 129 \times 128 \times 127 \times 126}{5 \times 4 \times 3 \times 2 \times 1} = 2,790,504 \].Using complementary counting, subtract the number of all-conforming samples from the total number of samples:\[ 4,479,848 - 2,790,504 = 1,689,344 \].

Key Concepts

Combination Formula

In the realm of statistics, the combination formula is a crucial tool for calculating how many different ways you can choose a specific number of items from a larger set, without regard to the order of selection. This is particularly useful in problems like the one involving semiconductor chips.

The formula itself is defined as \( \binom{n}{k} \), where:

• \( n \) is the total number of items to choose from.
• \( k \) is the number of items you wish to choose.

The formula is expressed as:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

For example, in the exercise with 140 total chips and selecting 5, you use \( \binom{140}{5} \). The calculation involves finding the factorial of numbers, which is a product of all integers up to a specific number. Utilizing this formula enables you to compute large datasets efficiently without physically counting every potential combination.

Nonconforming Samples

Nonconforming samples refer to those that contain items which do not meet certain criteria or standards, such as customer requirements in the context of manufacturing. Identifying how many samples include exactly one nonconforming chip involves strategic application of combinatorics.

In the case of semiconductor chips, you're tasked with selecting 5 chips, where exactly one among them doesn't conform. The process requires:

• Choosing 1 out of 10 nonconforming chips: \( \binom{10}{1} = 10 \).
• Selecting 4 conforming chips from the 130 that meet the criteria: \( \binom{130}{4} \).

By calculating both parts and multiplying them together, you pinpoint the exact number of viable sample configurations that contain exactly one nonconforming chip. This kind of detailed analysis is critical in quality control and predicting potential issues in production processes.

Complementary Counting

Complementary counting is an elegant technique in combinatorics, often used when determining the number of ways an event can happen is more complicated than calculating its complement.

In simpler terms, it integrates the idea of finding "what you don't want" to deduce "what you do want." For instance, if you're interested in at least one nonconforming chip in a sample of five, it's easier to count how many ways you can pick only conforming chips and subtract that from the total.

For the chip scenario, you:

• First calculate samples with only conforming chips: \( \binom{130}{5} \).
• Subtract that result from the total number of all possible samples: \( \binom{140}{5} \).

This approach reduces complexity and, upon subtraction, gives you the count of samples with at least one nonconforming chip—simplifying what could otherwise be a daunting combinatorial task. Complementary counting is essential in tackling problems where direct computation of each scenario is cumbersome.