Binomial Probability: Calculate P(2) With N=50, P=0.04

October 16, 2025 by TextBrain Team 55 views

Alright, let's dive into the world of binomial probability! In this article, we're going to break down how to calculate the probability of a specific number of successes in a set number of trials. Specifically, we'll tackle a problem where we need to find P(2), which represents the probability of getting exactly 2 successes in a binomial experiment. We'll be working with an experiment that has 50 trials (n = 50), and the probability of success on each trial is 0.04 (p = 0.04). So, buckle up, and let's get started!

Understanding Binomial Probability

Before we jump into the calculation, let's make sure we're all on the same page about what binomial probability is. Imagine you're flipping a coin multiple times, or maybe you're checking if a product is defective in a production line. These scenarios, where you have a fixed number of independent trials, each with two possible outcomes (success or failure), are perfect examples of situations where binomial probability comes into play. The key characteristics of a binomial experiment are:

A fixed number of trials (n).
Each trial is independent of the others.
There are only two possible outcomes: success or failure.
The probability of success (p) remains constant from trial to trial.

In our case, we have n = 50 trials, and the probability of success (p) is 0.04. We want to find the probability of getting exactly 2 successes (x = 2). The binomial probability formula helps us calculate this. Let's take a closer look at the formula itself and understand each of its components, so we are crystal clear on how to utilize it effectively.

The Binomial Probability Formula

The binomial probability formula might look a little intimidating at first, but don't worry, we'll break it down piece by piece. The formula is:

P(x) = (nCx) * p^x * (1 - p)^(n - x)

Where:

P(x) is the probability of getting exactly x successes.
n is the number of trials.
x is the number of successes we want to find the probability for.
p is the probability of success on a single trial.
(nCx) is the number of combinations of n items taken x at a time, also known as the binomial coefficient. It represents the number of ways to choose x successes from n trials.

Now, let's break down each part of the formula to understand it better. This formula is the cornerstone of our calculation, and having a firm grasp of each component will make the entire process much smoother and more intuitive for you, guys.

Understanding the Components

(nCx) - The Binomial Coefficient: This part might look a bit strange if you haven't seen it before. It's calculated as:
```
nCx = n! / (x! * (n - x)!)
```
Where "!" denotes the factorial, meaning the product of all positive integers up to that number (e.g., 5! = 5 * 4 * 3 * 2 * 1). The binomial coefficient tells us how many different ways we can get x successes in n trials. For example, if we are picking 2 winning tickets out of a lottery of 5 tickets, this part of the formula will tell us how many unique combinations of tickets can win. Knowing this helps put the overall probability in perspective.
p^x - Probability of x Successes: This is simply the probability of success on a single trial (p) raised to the power of the number of successes we want (x). It calculates the likelihood of getting x successes in a row. It's important because it emphasizes how each individual success contributes to the overall probability, especially in scenarios where the success rate is low.
(1 - p)^(n - x) - Probability of (n - x) Failures: This part calculates the probability of getting (n - x) failures. (1 - p) is the probability of failure on a single trial, and we raise it to the power of (n - x) to find the probability of getting that many failures. This is crucial because failures are just as much a part of the binomial distribution as successes. Without accounting for failures, our probability calculation would be incomplete.

By understanding each component, we can appreciate how the binomial probability formula works as a whole. It combines the number of ways to achieve a specific outcome with the probabilities of success and failure to give us the overall probability of that outcome. Now that we've got a handle on the formula, let's apply it to our problem.

Applying the Formula to Our Problem

Okay, now that we have a good understanding of the binomial probability formula, let's plug in the values from our problem. Remember, we have:

n = 50 (number of trials)
p = 0.04 (probability of success)
x = 2 (number of successes)

We want to find P(2), the probability of getting exactly 2 successes. Let's break this down step-by-step so you can see exactly how it works.

Step 1: Calculate the Binomial Coefficient (50C2)

First, we need to calculate 50C2, which is the number of ways to choose 2 successes from 50 trials. Using the formula:

50C2 = 50! / (2! * (50 - 2)!)

This looks intimidating, but let's simplify it:

50C2 = 50! / (2! * 48!)
     = (50 * 49 * 48!) / (2 * 1 * 48!)
     = (50 * 49) / 2
     = 1225

So, there are 1225 different ways to get 2 successes in 50 trials.

Step 2: Calculate p^x

Next, we need to calculate p^x, which is the probability of success (0.04) raised to the power of the number of successes (2):

p^x = 0.04^2
    = 0.0016

This is the probability of getting two successes in a row.

Step 3: Calculate (1 - p)^(n - x)

Now, let's calculate (1 - p)^(n - x), which is the probability of failure (1 - 0.04 = 0.96) raised to the power of the number of failures (50 - 2 = 48):

(1 - p)^(n - x) = 0.96^48
              ≈ 0.14228

This is the probability of getting 48 failures in 50 trials.

Step 4: Plug Everything into the Formula

Finally, we can plug all the values we calculated into the binomial probability formula:

P(2) = (50C2) * p^2 * (1 - p)^48
     = 1225 * 0.0016 * 0.14228
     ≈ 0.2792

So, the probability of getting exactly 2 successes in our binomial experiment is approximately 0.2792, or about 27.92%.

Interpreting the Result

Now that we've calculated P(2), let's take a moment to understand what this number means. A probability of 0.2792 (or 27.92%) tells us that if we were to repeat this binomial experiment (50 trials with a success probability of 0.04) many times, we would expect to get exactly 2 successes in about 27.92% of those experiments. It's not a super high probability, but it's also not negligible. This kind of insight is valuable in various real-world scenarios, from quality control to predicting outcomes in sports or games of chance.

Real-World Applications

Binomial probability isn't just a theoretical concept; it has tons of practical applications. For instance:

Quality Control: Imagine a factory producing light bulbs. If the probability of a bulb being defective is 0.04, we can use binomial probability to calculate the likelihood of finding a certain number of defective bulbs in a batch of 50.
Marketing: If a marketing campaign has a 4% success rate (meaning 4% of people click on an ad), we can use binomial probability to determine the probability of getting a specific number of clicks from a group of 50 people.
Medical Research: In clinical trials, binomial probability can be used to assess the effectiveness of a treatment. If a treatment has a 4% chance of causing side effects, we can calculate the probability of observing side effects in a group of 50 patients.

Understanding binomial probability allows us to make informed decisions and predictions in a wide range of fields. It's a powerful tool for analyzing situations where outcomes are binary (success or failure) and the probability of success is constant.

Tips for Calculating Binomial Probability

Calculating binomial probabilities can sometimes be a bit tricky, especially when dealing with large numbers. Here are some tips to help you avoid common mistakes and make the process smoother:

Double-Check Your Values: Make sure you've correctly identified n, p, and x. A small mistake here can throw off your entire calculation.
Use a Calculator or Software: Calculating factorials and large exponents by hand can be tedious and error-prone. Use a scientific calculator or software like Excel or R to do the heavy lifting.
Break It Down: If the formula seems overwhelming, break it down into smaller parts, as we did in this article. Calculate each component separately and then combine them.
Understand the Context: Always think about what the result means in the context of the problem. Does the probability you calculated make sense? If not, review your steps.
Practice, Practice, Practice: The more you work with binomial probability, the more comfortable you'll become with it. Try solving different problems with varying values of n, p, and x.

By following these tips, you'll be well on your way to mastering binomial probability calculations.

Conclusion

So, there you have it! We've walked through how to calculate binomial probability, specifically P(2) for an experiment with n = 50 and p = 0.04. We broke down the formula, calculated each component, and put it all together to get our answer. We also discussed the importance of understanding binomial probability and its many real-world applications. Hopefully, you now have a solid grasp of this concept and feel confident in tackling similar problems. Remember, practice makes perfect, so keep working at it, and you'll become a binomial probability pro in no time!

Keep practicing and exploring, guys, and you'll find that binomial probability is a powerful tool for understanding and predicting outcomes in a variety of situations. Whether you're analyzing data, making decisions, or simply trying to understand the world around you, the principles of binomial probability can provide valuable insights. Until next time, keep those calculations coming!