Problem 217 - Applied Statistics and Probability for Engineers

Question

A lot of 50 spacing washers contains 30 washers that are thicker than the target dimension. Suppose that 3 washers are selected at random, without replacement, from the lot. (a) What is the probability that all 3 washers are thicker than the target? (b) What is the probability that the third washer selected is thicker than the target if the first 2 washers selected are thinner than the target? (c) What is the probability that the third washer selected is thicker than the target?

Step-by-Step Solution

Answer

(a) 0.345, (b) 0.625, (c) ≈ 0.345

Step 1: Understand the Problem

We have a total of 50 washers. Out of them, 30 washers are thicker than the target dimension, and 20 are thinner. We want to find probabilities for different scenarios involving selecting 3 washers at random without replacement.

Step 2: Calculate Probability for Scenario (a)

To find the probability that all 3 washers are thicker than the target, we calculate the probability of picking thick washers in succession without replacement. First washer is thick: \( \frac{30}{50} \) Second washer (thick) given first is thick: \( \frac{29}{49} \) Third washer (thick) given first two are thick: \( \frac{28}{48} \) Multiplying these probabilities together gives us: \[ P(a) = \frac{30}{50} \times \frac{29}{49} \times \frac{28}{48} = \frac{4060}{11760} = \frac{203}{588} \approx 0.345 \]

Step 3: Calculate Probability for Scenario (b)

For scenario (b), if the first two washers are thin, we have a different situation. Probability first washer is thin: \( \frac{20}{50} \)Probability second washer is thin given the first was thin: \( \frac{19}{49} \)Now, there are 30 thick washers left out of 48 total: \[ P(b) = \frac{20}{50} \times \frac{19}{49} \times \frac{30}{48} \]Focus on the third washer: \[ P(\, \text{third thick} \mid \text{first two thin}) = \frac{30}{48} = \frac{5}{8} \approx 0.625 \]

Step 4: Calculate Probability for Scenario (c)

We want to consider all sequences where the third washer is thick. We account for all possible preceding selections: TTT, TTT, and so on. An easier method is to reverse the problem.Consider selecting 2 washers, and we need at least one thick washer in the three positions. Using the complementary approach:Probability all thin in first 2 drawn: \[ P(\text{2 thin}) = \frac{20}{50} \times \frac{19}{49} \]Probability last (3rd) washer is thick is:\[ P(\text{any thick}) = 1 - P(\text{all 3 thin}) \approx 0.345 \]

Step 5: Review and Verification

Double-check calculations for errors, ensuring no computational or logical errors are present. Verify each scenario is distinct and considered independently.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and selecting objects. It becomes particularly handy in probability when determining the number of ways events can occur. In our exercise, we applied combinatorial thinking when assessing how washers are selected without replacement from a total lot of 50 washers.

When considering probabilities, we need to assess all possible outcomes. For example, in the situation where we want to determine the likelihood of drawing three thick washers in a row, we have 50 washers to start, then 49 after removing one, and so forth. The order we draw these washers matters in this case because we're not replacing each washer drawn.

Combinatorics provides powerful tools such as permutations and combinations. These allow us to evaluate different scenarios without having to list each one manually. For events where order matters, permutations are the tool of choice. In our instance, where specific order doesn't matter, we often lean on combinations. This focus allows us to hone in directly on the probability with precision and ease.

Conditional Probability

Conditional probability deals with the probability of an event happening given that another event has already occurred. This concept is crucial in many real-world scenarios where past events affect future outcomes.

In our problem, part (b) required us to find the probability that the third washer selected is thicker than the target, given that the first two washers were thinner. Here, the condition alters the total number of washers that are eligible as thick or thin. Initially, we might think that each selection is independent. However, the choosing process without replacement directly alters the probabilities at each stage, which is at the heart of conditional probability.

• Initial condition: The first two washers are thin.
• Outcome to evaluate: The third washer is thick.

By factoring in these conditions, we refine our probability calculation to be more precise. This refinement is reflected in our probability formula for the third washer, ensuring we aren't overestimating or underestimating based on unfounded assumptions. Thus, conditional probability allows for a deeper, more accurate understanding of probabilities in step-by-step dependent scenarios.

Probability Theory

Probability theory is the mathematical framework we use to quantify and manage uncertainty. It is a universal language for discussing how likely events are to occur. In our washer example, probability theory helps us model the likelihood of different outcomes when washers are drawn sequentially from a set.

The entire exercise hinges on understanding and applying the foundational rules of probability, such as the rule of product (multiplying successive events' probabilities) and using complementary probabilities. For question (c), we considered how to assess the likelihood of drawing at least one thick washer in three draws. Here, we employed the concept of complementary probability, which entails considering the probability of the opposite occurring (all washers being thin) and subtracting from one.

• Problem-solving method: Exploiting complements to simplify potentially complex calculations.
• Probability basics: Understanding independence and dependence in a non-replacement scenario.

It's this robust nature of probability theory that allows us to tackle questions that might otherwise feel nebulous and uncertain. With probability theory, we can translate real-life random processes into calculable events, making it an invaluable tool for a wide range of disciplines.