Calculating Binomial Probability Exactly Three Successes In Five Trials

Hey guys! Let's dive into a classic probability problem, specifically dealing with binomial experiments. These experiments are super common and pop up in all sorts of scenarios, from coin flips to quality control in manufacturing. Today, we're tackling a question that asks us to calculate a specific binomial probability. We'll break it down step-by-step, so you'll not only get the answer but also understand the why behind it.

The Problem: Exactly Three Successes

Our mission, should we choose to accept it (and we do!), is to find the probability of exactly three successes in a binomial experiment. This experiment consists of five independent trials, and the probability of success on any single trial is a whopping 90%. We need to express our final answer as a percentage, rounded to the nearest tenth of a percent. So, let's put on our probability hats and get started!

Decoding Binomial Experiments

Before we jump into the calculations, let's make sure we're all on the same page about what a binomial experiment actually is. Think of it as a series of identical trials, each with only two possible outcomes: success or failure. The key characteristics that define a binomial experiment are:

• Fixed Number of Trials: We know exactly how many times the experiment will be repeated (in our case, five trials).
• Independent Trials: The outcome of one trial doesn't affect the outcome of any other trial. Like flipping a coin – one flip doesn't influence the next.
• Two Possible Outcomes: Each trial results in either success or failure (think heads or tails, pass or fail, etc.).
• Constant Probability of Success: The probability of success remains the same for every trial (our 90% success rate).

These characteristics are crucial because they allow us to use the binomial probability formula, which is the magic key to solving our problem.

The Binomial Probability Formula: Our Secret Weapon

Here it is, the formula that will help us conquer this probability puzzle:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

Don't let the formula intimidate you! Let's break it down piece by piece:

• P(X = k): This is what we're trying to find – the probability of getting exactly k successes.
• (n choose k): This is the binomial coefficient, also known as "n choose k." It represents the number of ways to choose k successes from n trials. We calculate it as n! / (k! * (n - k)!). The exclamation mark means factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1).
• p: This is the probability of success on a single trial (our 90%, or 0.9).
• k: This is the number of successes we're interested in (exactly three, in our problem).
• (1 - p): This is the probability of failure on a single trial (1 - 0.9 = 0.1).
• n: This is the total number of trials (five, in our case).

With the formula in hand and each component defined, we're ready to plug in our values and calculate the probability.

Plugging in the Values: Let's Get Calculating!

Now comes the fun part – substituting our specific values into the binomial probability formula. Remember, we want to find the probability of exactly three successes (k = 3) in five trials (n = 5) with a success probability of 90% (p = 0.9). Let's do it!

P(X = 3) = (5 choose 3) * (0.9)^3 * (0.1)^(5 - 3)

Let's tackle each part of the equation step-by-step:

Calculating the Binomial Coefficient

First up, the binomial coefficient: (5 choose 3). This tells us how many different ways we can get three successes in five trials. Using the formula:

(5 choose 3) = 5! / (3! * 2!) = (5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (2 * 1)) = 120 / (6 * 2) = 10

So, there are 10 different ways to get exactly three successes in five trials.

Calculating the Probability of Successes

Next, we need to calculate (0.9)^3, which is the probability of getting three successes in a row:

(0. 9)^3 = 0.9 * 0.9 * 0.9 = 0.729

Calculating the Probability of Failures

Now, let's calculate (0.1)^(5 - 3) = (0.1)^2, which is the probability of getting two failures:

(0.1)^2 = 0.1 * 0.1 = 0.01

Putting It All Together

Finally, we can plug all these values back into our formula:

P(X = 3) = 10 * 0.729 * 0.01 = 0.0729

So, the probability of getting exactly three successes in five trials is 0.0729. But remember, the question asks for the answer as a percentage, rounded to the nearest tenth.

Converting to Percentage and Rounding

To convert the probability to a percentage, we simply multiply by 100:

0. 0729 * 100 = 7.29%

Now, rounding to the nearest tenth of a percent, we get:

7. 3%

The Answer: Success! (Pun Intended)

Therefore, the probability of exactly three successes in five trials of this binomial experiment is approximately 7.3%. We did it!

Key Takeaways and Real-World Applications

This problem highlights the power of the binomial probability formula in calculating the likelihood of specific outcomes in situations with repeated independent trials. Understanding binomial probabilities is crucial in various fields, including:

• Quality Control: Assessing the probability of defective items in a production batch.
• Medical Research: Determining the effectiveness of a new drug or treatment.
• Market Research: Estimating the proportion of consumers who prefer a certain product.
• Genetics: Predicting the probability of inheriting specific traits.

The binomial distribution is a fundamental concept in probability and statistics, and mastering it will open doors to understanding a wide range of real-world phenomena.

Practice Makes Perfect: More Binomial Fun

Now that you've tackled this problem, try applying the binomial probability formula to other scenarios. What if the probability of success was different? What if we had a different number of trials? Playing around with these variables will solidify your understanding and make you a binomial probability pro! You got this! Remember to always break down the problem, identify the key components, and apply the formula step-by-step. Happy calculating!