Probability Of Coffee Given Small Size: Step-by-Step Guide

Hey guys! Let's dive into a probability problem that might seem a bit tricky at first, but trust me, we'll break it down together. We're going to figure out the probability that a drink is coffee, given that it's a small size. To do this, we'll use a handy table and a little bit of conditional probability magic. So, grab your thinking caps, and let's get started!

Understanding Conditional Probability

Before we jump into the specific problem, let's make sure we're all on the same page about conditional probability. Conditional probability is all about finding the probability of an event happening, knowing that another event has already occurred. Think of it like this: we're narrowing down our focus to a specific situation. In our case, we already know the drink is small, and we want to find the probability that it's also coffee. The formula for conditional probability is:

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

Where:

• P(A|B) is the probability of event A happening given that event B has happened.
• P(A ∩ B) is the probability of both events A and B happening.
• P(B) is the probability of event B happening.

In simpler terms, it's the probability of both events happening divided by the probability of the given event. Now that we've got the formula down, let's see how it applies to our coffee and small drinks!

Analyzing the Data Table

Let's take a look at the table we've been given. It's super helpful because it organizes all the information we need:

This table breaks down the drinks by size (Large and Small) and type (Coffee and Tea). The totals help us see the bigger picture. For example, we can see that there are 15 coffees in total, 20 small drinks in total, and 33 drinks overall. These numbers are going to be our building blocks for calculating the probability. To make things crystal clear, let's define our events:

• Event A: The drink is coffee.
• Event B: The drink is small.

Our goal is to find P(Coffee | Small), which means the probability that the drink is coffee given that it's small. We'll use the numbers from the table to plug into our conditional probability formula. First, we need to find P(Coffee ∩ Small), which is the probability that the drink is both coffee and small. We can find this directly from the table. Next, we'll find P(Small), the probability that the drink is small. With these two pieces of information, we'll be able to calculate our final answer!

Calculating the Probabilities

Okay, let's crunch some numbers! We need to find two key probabilities:

1. P(Coffee ∩ Small): This is the probability that a drink is both coffee AND small. Looking at the table, we can see that there are 7 drinks that fit this description. There are a total of 33 drinks, so:

   $$P(Coffee \cap Small) = \frac{7}{33}$$

   So, the probability of picking a drink that is both coffee and small is 7 out of 33. This is our numerator in the conditional probability formula.

2. P(Small): This is the probability that a drink is small. The table tells us there are 20 small drinks out of a total of 33 drinks. Therefore:

   $$P(Small) = \frac{20}{33}$$

   This means that if you were to randomly pick a drink, there's a 20 out of 33 chance it would be small. This is our denominator in the conditional probability formula. Now that we have both probabilities, we're ready to plug them into the formula and find our answer!

Applying the Conditional Probability Formula

Alright, we've got all the pieces of the puzzle! Let's put them together using the conditional probability formula:

$$P(Coffee | Small) = \frac{P(Coffee \cap Small)}{P(Small)}$$

We already calculated:

• P(Coffee ∩ Small) = 7/33
• P(Small) = 20/33

Now, let's substitute these values into the formula:

$$P(Coffee | Small) = \frac{\frac{7}{33}}{\frac{20}{33}}$$

To simplify this, we can multiply the numerator by the reciprocal of the denominator:

$$P(Coffee | Small) = \frac{7}{33} \times \frac{33}{20}$$

The 33s cancel out, leaving us with:

$$P(Coffee | Small) = \frac{7}{20}$$

So, the probability that a drink is coffee, given that it is small, is 7/20. This means that out of all the small drinks, 7 out of 20 of them are coffee. We've successfully navigated through the problem and found our answer! Remember, conditional probability is all about narrowing your focus based on new information, and we did just that by considering only the small drinks.

Converting to Percentage (Optional)

Sometimes, it's helpful to express probabilities as percentages. To convert 7/20 to a percentage, we simply divide 7 by 20 and then multiply by 100:

$$\frac{7}{20} = 0.35$$

$$0.35 \times 100 = 35\%$$

So, we can also say that there is a 35% chance that a drink is coffee, given that it is small. This can make the result a bit more intuitive for some people. Whether you prefer the fraction or the percentage, the key is that we've accurately calculated the conditional probability.

Real-World Applications of Conditional Probability

You might be wondering,