Probability Of Defective Toys A Step By Step Guide

Hey there, math enthusiasts! Let's dive into a fascinating probability problem that involves Mrs. Jones and her quest to buy toys for her son. We'll break down the problem step by step, making sure everyone understands the concepts involved. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving the problem, let's make sure we understand it completely. Mrs. Jones is buying two toys for her son. The problem gives us two crucial pieces of information:

1. The probability that the first toy is defective is $rac{1}{3}$.
2. The probability that the second toy is defective, given that the first toy is defective, is $rac{1}{5}$.

Our goal is to figure out the probability that both toys are defective. This is a classic example of a conditional probability problem, where the outcome of the first event (the first toy being defective) affects the probability of the second event (the second toy being defective).

Key Concepts in Probability

To tackle this problem effectively, let's quickly review some key concepts in probability:

• Probability: The likelihood of an event occurring, expressed as a number between 0 and 1. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain.
• Independent Events: Events where the outcome of one event does not affect the outcome of the other. For example, flipping a coin twice. The result of the first flip doesn't change the probability of the second flip.
• Dependent Events: Events where the outcome of one event does affect the outcome of the other. Our toy problem is an example of dependent events because the condition of the first toy influences the probability of the second toy being defective.
• Conditional Probability: The probability of an event occurring given that another event has already occurred. We denote the conditional probability of event B given event A as P(B|A).
• Joint Probability: The probability of two events both occurring. The joint probability of events A and B is denoted as P(A and B).

Conditional Probability Formula

The key to solving this problem lies in understanding the formula for conditional probability:

$$P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$$

Where:

• P(B|A) is the probability of event B occurring given that event A has occurred.
• P(A and B) is the probability of both events A and B occurring.
• P(A) is the probability of event A occurring.

Solving the Problem

Now that we have the necessary background, let's apply these concepts to our toy problem.

Defining the Events

First, let's define our events:

• Event A: The first toy is defective.
• Event B: The second toy is defective.

We are given:

• P(A) = $rac{1}{3}$ (The probability that the first toy is defective)
• P(B|A) = $rac{1}{5}$ (The probability that the second toy is defective given that the first toy is defective)

Finding the Joint Probability

We want to find the probability that both toys are defective, which is P(A and B). To do this, we can rearrange the conditional probability formula:

$$P(A \text{ and } B) = P(B|A) \times P(A)$$

Now, we can plug in the values we know:

$$P(A \text{ and } B) = \frac{1}{5} \times \frac{1}{3}$$

Calculating the Result

Multiplying the fractions, we get:

$$P(A \text{ and } B) = \frac{1 \times 1}{5 \times 3} = \frac{1}{15}$$

Therefore, the probability that both toys are defective is $rac{1}{15}$.

Step-by-Step Breakdown

Let's recap the steps we took to solve this problem:

1. Understand the Problem: We carefully read and understood the information given in the problem statement.
2. Identify Key Concepts: We recognized that this was a conditional probability problem and reviewed relevant concepts.
3. Define Events: We clearly defined the events A and B.
4. Apply the Formula: We used the conditional probability formula to find the joint probability.
5. Calculate the Result: We performed the necessary calculations to arrive at the final answer.

Why is this important?

Understanding conditional probability isn't just about solving math problems; it's a valuable skill in many areas of life. For example:

• Decision Making: Conditional probability helps us make informed decisions by considering how past events influence future probabilities. Think about medical diagnoses, risk assessments, or even business strategies.
• Data Analysis: In statistics and data science, conditional probability is used to analyze relationships between variables and make predictions.
• Everyday Life: We use conditional probability intuitively all the time. For instance, if it's cloudy, we might think there's a higher probability of rain.

Another example

Let's consider another example to solidify your understanding. Suppose a factory produces light bulbs. The probability that a bulb is defective is 0.05. If a bulb is defective, the probability that it will shatter during shipping is 0.2. What is the probability that a bulb is both defective and shatters during shipping?

Let's define our events:

• Event D: The bulb is defective.
• Event S: The bulb shatters during shipping.

We are given:

• P(D) = 0.05
• P(S|D) = 0.2

We want to find P(D and S). Using the conditional probability formula:

$$P(D \text{ and } S) = P(S|D) \times P(D)$$

Plugging in the values:

$$P(D \text{ and } S) = 0.2 \times 0.05 = 0.01$$

So, the probability that a bulb is both defective and shatters during shipping is 0.01.

Conclusion: Mastering Probability

Probability problems, especially those involving conditional probability, can seem tricky at first. But with a clear understanding of the concepts and a step-by-step approach, you can tackle them with confidence. Remember to define your events, identify the relevant probabilities, and apply the appropriate formulas. Keep practicing, and you'll become a probability pro in no time!

So, next time you encounter a problem like Mrs. Jones's toy dilemma, you'll be ready to solve it like a champ. Keep exploring the world of probability, guys, it's full of fascinating insights!