Question

A shipping container contains seven complex electronic systems. Unknown to the purchaser, three are defective. Two of the seven are selected for thorough testing and are then classified as defective or nondefective. What is the probability that no defectives are found?

Short answer

Answer: The probability is 2/7.

Step-by-step solution

Determine the total number of combinations

To find the total number of ways to choose two systems out of seven for testing, we will use the combination formula: C(n, k) = n! / (k!(n-k)!) where C(n, k) is the number of ways to choose k items from a set of n items, n! is the factorial of n, and k! is the factorial of k. In our case, n = 7 (total electronic systems) and k = 2 (systems selected for testing). C(7, 2) = 7! / (2! * (7-2)!) C(7, 2) = 7! / (2! * 5!) C(7, 2) = 5040 / (2 * 120) C(7, 2) = 5040 / 240 C(7, 2) = 21 There are 21 different combinations of testing two electronic systems out of seven.

Determine how many combinations do not have any defective systems

We have four non-defective systems and we need to find the number of combinations of choosing two non-defective systems for testing. Apply the same combination formula as in step 1 with n = 4 (total non-defective systems) and k = 2. C(4, 2) = 4! / (2! * (4-2)!) C(4, 2) = 4! / (2! * 2!) C(4, 2) = 24 / (2 * 2) C(4, 2) = 24 / 4 C(4, 2) = 6 There are 6 combinations of testing two non-defective electronic systems.

Compute the probability

To find the probability that no defective systems are found, we will divide the number of combinations without defective systems by the total number of combinations: Probability = (Number of combinations without defective systems) / (Total number of combinations) Probability = 6 / 21 Probability = 2 / 7 The probability that no defective systems are found when two systems are selected for thorough testing is 2/7.

Key concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. It provides essential tools for solving problems related to probability.

The basics of combinatorics involve understanding permutations and combinations. While permutations concern arrangements where order matters, combinations focus on selections where order does not matter.

In the exercise, we used combinations to select systems from a batch, relying on the formula \( C(n, k) = \frac{n!}{k!(n-k)!} \). Here, \( n \) represents the total number of objects, and \( k \) is the number of objects to choose. This method allows us to calculate how many ways we can select systems, such as our seven systems, two at a time, without considering order.

By using combinations, we systematically explore every possible selection scenario, a key step when assessing probabilities in complex situations.

Defective Systems

Defective systems in this context refer to a subset of objects that do not meet the required quality standards. Identifying and calculating the probability of selecting these defective items is crucial in various fields, such as manufacturing and quality control.

In our problem, the container holds seven systems, out of which three are faulty. This scenario reflects a common issue: understanding both the likelihood of encountering defects and the impact of such on system performance.

To solve the problem, we specifically calculated how many ways we can choose only non-defective systems out of the total mix. Understanding this mix of defectives versus non-defectives aids in making informed decisions on quality assurance and risk management.

Probability Calculation

Probability calculation helps quantify the chance of a particular outcome occurring. It's a fundamental concept in probability theory, necessary for predicting the likelihood of various events.

To find probabilities, we use the ratio of favorable outcomes (events of interest) to the total number of possible outcomes. In the exercise, we sought the chance of picking two non-defective systems from seven, which involved dividing the number of favorable combinations by the total combinations.

Here's a simple breakdown:

• The total possible combinations of picking two systems among seven is 21, as determined using combinatorics.
• The combinations featuring only non-defective systems were calculated to be 6.
• Thus, the probability that no defectives are found is \( \frac{6}{21} \) or \( \frac{2}{7} \).

This straightforward ratio calculation illustrates the power of probability to evaluate and anticipate outcomes in decision-making and forecasting scenarios.