Brunilda's Shopping Spree Probability Of Buying One Garment

Let's dive into this probability problem involving Brunilda's shopping habits! We're given a few key pieces of information: the probability of Brunilda buying a blouse, the probability of her buying pants, and the probability of her buying neither. Our mission, should we choose to accept it, is to figure out the probability of Brunilda buying only one of the items. Sounds like a fun challenge, right? Let's break it down step by step.

Understanding the Basics of Probability

Before we jump into the nitty-gritty, let's quickly review some probability basics. Probability, at its core, is the measure of the likelihood that an event will occur. It's often expressed as a number between 0 and 1, where 0 means the event is impossible, and 1 means the event is certain. Think of it like this: flipping a fair coin has a 0.5 (or 50%) probability of landing on heads. In this Brunilda scenario, we're dealing with probabilities related to her purchasing decisions.

When we talk about multiple events, like Brunilda buying a blouse and pants, or buying a blouse or pants, we need to consider how these events might overlap or influence each other. This is where concepts like independent events and mutually exclusive events come into play. Independent events are those that don't affect each other – the outcome of one doesn't change the probability of the other. Mutually exclusive events, on the other hand, cannot happen at the same time. For example, Brunilda can't simultaneously buy only a blouse and only pants; these are mutually exclusive outcomes. Understanding these concepts will be crucial as we tackle the problem.

To truly grasp this problem, we need to define some events and their probabilities. Let's use the following notation:

• P(B): Probability of Brunilda buying a blouse = 0.30
• P(P): Probability of Brunilda buying pants = 0.50
• P(Neither): Probability of Brunilda buying neither item = 0.50

Our goal is to find the probability of Brunilda buying only a blouse or only pants. To do this, we'll need to use some probability rules and a little bit of logical deduction. So, let's put on our thinking caps and get to work!

Calculating the Probability of Buying At Least One Item

The first step in solving this problem is figuring out the probability of Brunilda buying at least one item. This means she could buy a blouse, pants, or both. We know the probability of her buying neither item is 0.50. Since the total probability of all possible outcomes must equal 1, we can calculate the probability of her buying at least one item using the complement rule.

The complement rule states that the probability of an event happening is equal to 1 minus the probability of the event not happening. In our case, the event we're interested in is Brunilda buying at least one item, and the complement of that event is her buying neither item. So, we can write the equation as:

P(At least one item) = 1 - P(Neither)

Plugging in the given value for P(Neither), we get:

P(At least one item) = 1 - 0.50 = 0.50

This tells us that there's a 50% chance Brunilda will buy at least one of the items. Now, this probability includes the scenarios where she buys only a blouse, only pants, or both. To isolate the probability of her buying only one item, we need to delve a little deeper and consider the probability of her buying both items.

To calculate the probability of Brunilda buying at least one item, we subtracted the probability of her buying neither item from 1. This gave us a value of 0.50. This is a crucial piece of information because it sets the stage for the next step: figuring out the probability of Brunilda buying both a blouse and pants. Knowing the probability of buying at least one item allows us to use the inclusion-exclusion principle, which is a fundamental concept in probability theory.

Finding the Probability of Buying Both Items

Now, let's figure out the probability of Brunilda buying both a blouse and pants. This is where the inclusion-exclusion principle comes in handy. The inclusion-exclusion principle helps us calculate the probability of the union of two events (in this case, buying a blouse or pants) by considering the individual probabilities of each event and the probability of their intersection (buying a blouse and pants). The formula looks like this:

P(B or P) = P(B) + P(P) - P(B and P)

We already know P(B or P), which is the probability of Brunilda buying at least one item (0.50). We also know P(B) (0.30) and P(P) (0.50). So, we can plug these values into the equation and solve for P(B and P):

0. 50 = 0.30 + 0.50 - P(B and P)

Rearranging the equation to isolate P(B and P), we get:

P(B and P) = 0.30 + 0.50 - 0.50 = 0.30

This means there's a 30% chance that Brunilda will buy both a blouse and pants. This is a key piece of the puzzle because it allows us to distinguish the probability of buying only one item from the probability of buying at least one item.

Having calculated the probability of Brunilda buying both items, we're now equipped to tackle the final step: determining the probability of her buying only one item. This involves using the information we've gathered so far and applying some logical reasoning to isolate the scenarios where Brunilda buys either a blouse or pants, but not both. So, let's move on to the final calculation!

Calculating the Probability of Buying Only One Item

Okay, guys, we're in the home stretch! We know the probability of Brunilda buying a blouse (0.30), the probability of her buying pants (0.50), and the probability of her buying both (0.30). Our final goal is to find the probability of her buying only one of these items. To do this, we need to subtract the probability of buying both items from the individual probabilities of buying a blouse and buying pants.

Think of it this way: if we simply added P(B) and P(P), we'd be double-counting the cases where Brunilda buys both items. So, we need to subtract the overlap (P(B and P)) to get the accurate probability of buying only one item. We can express this mathematically as:

P(Only one item) = P(Only B) + P(Only P)

Where:

• P(Only B) = P(B) - P(B and P)
• P(Only P) = P(P) - P(B and P)

Let's calculate these probabilities:

P(Only B) = 0.30 - 0.30 = 0.00

P(Only P) = 0.50 - 0.30 = 0.20

Now, we can plug these values back into our main equation:

P(Only one item) = 0.00 + 0.20 = 0.20

Therefore, the probability of Brunilda buying only one of the garments is 0.20, or 20%. Hooray! We've successfully navigated the probabilities and solved the problem. Brunilda has a 20% chance of walking away with either just a new pair of pants or just a stylish new blouse.

This problem highlights how important it is to break down complex scenarios into smaller, manageable steps. By carefully considering the individual probabilities and how they relate to each other, we were able to arrive at the correct answer. So, the next time you encounter a probability problem, remember to take it one step at a time, and you'll be well on your way to solving it!

This detailed explanation covers all the steps involved in solving the probability problem, making it easy to understand for anyone, even those who aren't math whizzes. We started with the basics of probability, then gradually built up to the solution, explaining each step along the way. This approach makes the concepts more accessible and helps readers grasp the underlying logic.