Problem 207 - Applied Statistics and Probability for Engineers

Question

A steel plate contains 20 bolts. Assume that 5 bolts are not torqued to the proper limit. 4 bolts are selected at random, without replacement, to be checked for torque. (a) What is the probability that all 4 of the selected bolts are torqued to the proper limit? (b) What is the probability that at least 1 of the selected bolts is not torqued to the proper limit?

Step-by-Step Solution

Answer

(a) Probability all are torqued: approximately 0.281. (b) Probability at least one is not torqued: approximately 0.719.

Step 1: Understand the Problem

We have a steel plate with 20 bolts, out of which 5 are not torqued properly. We're selecting 4 bolts at random without replacement to check their torque. We need to find the probability of all bolts being torqued properly and the probability of at least one not being torqued properly.

Step 2: Calculate Total Ways to Select Bolts

The total number of ways to choose 4 bolts out of 20 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number and \( r \) is the number to choose. Thus, \( \binom{20}{4} = \frac{20!}{4!(20-4)!} = 4845 \).

Step 3: Calculate Ways to Select Properly Torqued Bolts

Out of 15 properly torqued bolts, we want to choose all 4. The number of ways is \( \binom{15}{4} = \frac{15!}{4!(15-4)!} = 1365 \).

Step 4: Calculate Probability of All Properly Torqued Bolts

The probability that all 4 selected bolts are properly torqued is the ratio of favorable outcomes to the total outcomes, \( P(\text{all torqued}) = \frac{\binom{15}{4}}{\binom{20}{4}} = \frac{1365}{4845} \approx 0.281 \).

Step 5: Use Complement to Find Second Probability

To find the probability that at least one bolt is not torqued properly, we use the complement of the previous probability: \( P(\text{at least 1 not torqued}) = 1 - P(\text{all torqued}) = 1 - 0.281 = 0.719 \).

Key Concepts

Combinatorics

Combinatorics is a fundamental branch of mathematics that focuses on counting, arranging, and probability. One of its main objectives is to determine how many ways we can arrange or choose things under certain conditions.
In the context of selecting bolts from a steel plate as in this problem, combinatorics helps us calculate the different possible selections we can make.
When choosing items like bolts, it's crucial to understand the concept of combinations, which are used when the order of selection doesn't matter.
This leads us to using a specific formula to help us determine the number of possible ways to make these choices. This formula is presented as the binomial coefficient, denoted by \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number we want to select. This helps simplify the complexity of counting manually possible arrangements or selections.

Binomial Coefficient

The binomial coefficient is a key tool in combinatorics used to calculate the number of ways to choose \( r \) elements from a set of \( n \) elements, without considering the order. It's represented mathematically as \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n! \) ("n factorial") is the product of all positive integers up to \( n \).
The exercise utilizes this concept by determining how many ways we can select 4 bolts from a total of 20, denoted as \( \binom{20}{4} \). This helps calculate the total possible combinations of bolt selections.
Moreover, when finding the number of ways to pick bolts that are properly torqued, \( \binom{15}{4} \) is used, because there are 15 torqued bolts in total. Understanding the binomial coefficient is fundamental to solving problems involving combinations and is extensively applied in probability and statistics.

Complement Rule

The complement rule in probability is a handy tool for solving problems where calculating the probability of an event directly might be complicated. Instead, it's often easier to calculate the probability of the "complement" event and then subtract it from 1.
The complement of an event is whatever other possibilities might occur apart from the event itself. For example, if we are interested in the probability that at least one bolt is not properly torqued, it's more efficient to first find the probability that all bolts are properly torqued.
Once we determine the probability that all selected bolts are properly torqued (approximately 0.281), we then subtract this from 1 to find the complement, which provides the probability of at least one bolt not being torqued properly. Using this rule simplifies calculations when dealing with probability in larger sample spaces or complex events.