Problem 12 - Technical Mathematics with Calculus

Question

On a certain production line, \(5 \%\) of the parts are underweight and \(8 \%\) are overweight. What is the probability that one part selected at random is either underweight or overweight?

Step-by-Step Solution

Answer

The probability that a part is either underweight or overweight is 0.13 or 13%.

Step 1: Understanding the Problem

We are given two different probabilities: the probability that a part is underweight is 5%, and the probability that a part is overweight is 8%. We need to find the probability that a part is either underweight or overweight. Since these two events are mutually exclusive (a part cannot be both underweight and overweight at the same time), we simply add the two probabilities together.

Step 2: Converting Percentages to Probabilities

Before we add the probabilities, we need to convert the given percentages into decimal form. To do this, we divide the percentages by 100. Therefore, the probability of a part being underweight is 0.05 and the probability of being overweight is 0.08.

Step 3: Adding the Probabilities

Since being underweight and being overweight are mutually exclusive events, we can add their probabilities to find the total probability of a part being either underweight or overweight. The sum of the probabilities is 0.05 (underweight) + 0.08 (overweight) = 0.13.

Key Concepts

Mutually Exclusive Events

When two or more events cannot occur at the same time, they are known as mutually exclusive events. In the context of probability, this means that the occurrence of one event excludes the possibility of the other event happening. For example, when flipping a coin, the events 'landing on heads' and 'landing on tails' are mutually exclusive because the coin cannot show both sides at once.

In our production line scenario, a part cannot be both underweight and overweight simultaneously; it must be one or the other if it deviates from the standard weight. Recognizing when events are mutually exclusive is critical for properly calculating the overall probability of these events occurring. By correctly identifying such exclusive events, it's possible to apply the probability addition rule without the risk of double-counting any outcomes, ensuring the accuracy of probability calculations.

Percentages to Probabilities

To work with probabilities in a mathematical context, it's important to convert percentages to decimal probabilities. This is because probabilities are expressed as numbers between 0 and 1, with 0 indicating an impossible event and 1 indicating a certain event. To transform a percentage into a probability, simply divide the percentage by 100. This moves the decimal point two places to the left. For example:

• A 5% chance of an event occurring becomes a probability of 0.05.
• An 8% chance is converted to 0.08 as a probability.

Understanding how to make this conversion is fundamental because it enables the use of probabilities in calculations, such as when applying the probability addition rule to determine the likelihood of different outcomes.

Probability Addition Rule

The probability addition rule is a fundamental concept in probability theory, particularly when calculating the probability of either one event or another occurring. However, this rule only applies when dealing with mutually exclusive events, where the two events cannot happen at the same time. The rule states that the probability of one or the other event occurring is the sum of the individual probabilities.

In our textbook example, the probability of a part being either underweight or overweight is calculated by adding the individual probabilities of each separate event, because these events are mutually exclusive:

• Probability of underweight: 0.05.
• Probability of overweight: 0.08.

By adding them, 0.05 + 0.08, we get 0.13. Thus, there is a 13% chance, or a probability of 0.13, that a part selected at random from the production line will be either underweight or overweight. It's important not to apply this rule to events that can occur simultaneously, as that would require a different approach to calculate the combined probability.