Question

You bought a new set of four tires from a manufacturer who just announced a recall because \(2 \%\) of those tires are defective. What is the probability that at least one of yours is defective?

Short answer

The probability that at least one tire is defective is approximately 7.7%.

Step-by-step solution

Understanding the problem

You have 4 tires and each has a 2% chance of being defective. We need to find the probability that at least one of these tires is defective.

Identify what 'at least one' means

The event 'at least one tire is defective' is the complement of the event 'none of the tires are defective'. This means one or more defective tires: 1 defective, 2 defective, 3 defective, or all 4 defective.

Calculate the probability of a single tire being non-defective

Since the probability of a tire being defective is 2% or 0.02, the probability that a single tire is not defective is 0.98 (which is 1 minus 0.02).

Calculate the probability that all four tires are non-defective

The probability that all four tires are non-defective can be found by multiplying the probability that one tire is non-defective by itself 4 times: \[ P( ext{none defective}) = 0.98^4 \]

Calculate the probability that at least one tire is defective

To find the probability that at least one tire is defective, we subtract the probability that none of the tires are defective from 1: \[ P( ext{at least one defective}) = 1 - P( ext{none defective}) = 1 - 0.98^4 \]

Compute the final probability

Now we compute using a calculator: \[ P( ext{at least one defective}) = 1 - 0.98^4 \approx 0.077 \]Thus, the probability that at least one of the tires is defective is approximately 0.077, or 7.7%.

Key concepts

Complement Rule

To comprehend the probability that at least one tire is defective, we must first grasp the concept of complements in probability. The complement of an event is essentially the opposite event. In our context, the event "at least one tire is defective" is the complement of the event "none of the tires are defective."

In simpler terms, if you know the probability of none of the tires being defective, then you can easily find the probability of at least one tire being defective. This is done using the formula:

• The probability of "at least one" = 1 - (The probability of "none")

This relationship is handy because calculating the probability of something not happening (none defective) is often simpler than calculating the probability of it happening in multiple possible ways (one or more defective).

So, use this complement rule to tackle problems involving the phrase "at least one," making complex probability calculations much easier.

Defective Products

Defective products, especially in manufacturing, denote items that do not meet the expected standards or specifications. In our tire problem, a defective tire is one that might not perform safely or efficiently.

Knowing that 2% of the tires are defective, we face uncertainty concerning the quality of those we purchased. This 2% figure represents a probability – the likelihood of any given tire being defective. Understanding this concept helps students and consumers alike predict, assess, and manage potential risks associated with defective products.

In cases where manufacturers announce recalls due to defects, as in the problem scenario, it becomes crucial to determine the potential impact on your specific purchase. Using probability theory helps in making informed decisions on whether additional actions, like returning or replacing defective tires, are necessary.

Probability Calculation

Calculating probabilities involves understanding the chance or likelihood of a particular event occurring. In our exercise, we calculated the probability that none of the tires are defective, as well as the probability of at least one tire being defective.

• First, we found the probability that a single tire is non-defective: 0.98
• Then, we found the combined probability that all four tires are non-defective by multiplying the individual probabilities: \(0.98 \times 0.98 \times 0.98 \times 0.98 = 0.98^4\)
• Finally, using the complement rule, we calculated: \[ P(\text{at least one defective}) = 1 - 0.98^4 \]

This probability calculation process allows us to predict outcomes in various scenarios where uncertainty and multiple possibilities exist. By mastering these calculations, students can develop skills applicable in other statistical and real-world contexts.

Multiplication Rule

The multiplication rule in probability helps us find the chance that multiple independent events will occur together. When dealing with independent events, the probability of all events occurring is the product of their individual probabilities. In our scenario, we wanted to find the likelihood that no tires are defective.

• Since each tire being non-defective is an independent event, we multiplied the probability of one tire not being defective (0.98) by itself four times.
• Thus the calculation was \(0.98^4\).

By using the multiplication rule, it becomes straightforward to compute the probability that all four tires are in good condition. Understanding this rule is essential for solving problems involving multiple independent events, significantly simplifying the process of probability determinations in everyday and complex situations alike.