Calculating Binomial Probability For N=10, P=0.45, X=8

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In the realm of probability and statistics, the binomial probability distribution holds a prominent position. It serves as a powerful tool for analyzing experiments where there are a fixed number of independent trials, each with only two possible outcomes: success or failure. Understanding and calculating binomial probabilities is crucial in various fields, from quality control in manufacturing to predicting election outcomes. This comprehensive guide will walk you through the process of calculating the probability of a specific number of successes in a binomial experiment, focusing on a scenario where we have n = 10 independent trials, the probability of success on each trial is p = 0.45, and we want to find the probability of exactly x = 8 successes. We will delve into the underlying formula, provide a step-by-step calculation, and offer insights into interpreting the result. By the end of this guide, you will have a solid grasp of how to calculate binomial probabilities and apply this knowledge to real-world situations.

The cornerstone of calculating binomial probabilities is the binomial probability formula itself. This formula elegantly captures the essence of the binomial distribution, allowing us to determine the probability of obtaining a specific number of successes in a fixed number of trials. The formula is expressed as follows:

P(x)=(nx)∗px∗(1−p)(n−x)P(x) = \binom{n}{x} * p^x * (1-p)^{(n-x)}

Where:

  • P(x) represents the probability of obtaining exactly x successes.
  • n denotes the total number of independent trials.
  • x is the desired number of successes.
  • p signifies the probability of success on a single trial.
  • (1-p) represents the probability of failure on a single trial.
  • (nx)\binom{n}{x} is the binomial coefficient, which calculates the number of ways to choose x successes from n trials. It is also denoted as "n choose x" and can be calculated as:

(nx)=n!x!(n−x)!\binom{n}{x} = \frac{n!}{x!(n-x)!}

Where:

  • n! (n factorial) is the product of all positive integers up to n. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120.

The binomial coefficient plays a crucial role in the formula by accounting for all the possible combinations of successes and failures that can result in x successes. The term p^x calculates the probability of getting x successes in a specific order, while (1-p)^(n-x) calculates the probability of getting (n-x) failures in the remaining trials. Multiplying these terms by the binomial coefficient gives us the overall probability of obtaining x successes in any order.

Now, let's apply the binomial probability formula to the specific scenario presented: n = 10, p = 0.45, and x = 8. This means we have 10 independent trials, the probability of success on each trial is 0.45, and we want to find the probability of exactly 8 successes.

Step 1: Calculate the Binomial Coefficient

First, we need to calculate the binomial coefficient (108)\binom{10}{8}, which represents the number of ways to choose 8 successes from 10 trials:

(108)=10!8!(10−8)!=10!8!2!=10∗9∗8∗7∗6∗5∗4∗3∗2∗1(8∗7∗6∗5∗4∗3∗2∗1)(2∗1)=10∗92∗1=45\binom{10}{8} = \frac{10!}{8!(10-8)!} = \frac{10!}{8!2!} = \frac{10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1}{(8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)(2 * 1)} = \frac{10 * 9}{2 * 1} = 45

So, there are 45 different ways to obtain 8 successes in 10 trials.

Step 2: Calculate p^x

Next, we calculate the probability of getting 8 successes, which is p^x = 0.45^8:

0.458≈0.000168169200.45^8 ≈ 0.00016816920

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

Now, we calculate the probability of getting (10-8) = 2 failures, which is (1-p)^(n-x) = (1-0.45)^(10-8) = 0.55^2:

0.552=0.30250.55^2 = 0.3025

Step 4: Plug the Values into the Formula

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

P(8)=(108)∗0.458∗0.552=45∗0.00016816920∗0.3025≈0.00229P(8) = \binom{10}{8} * 0.45^8 * 0.55^2 = 45 * 0.00016816920 * 0.3025 ≈ 0.00229

Therefore, the probability of obtaining exactly 8 successes in 10 independent trials, with a success probability of 0.45 on each trial, is approximately 0.00229.

To solidify your understanding, let's break down the calculation into a clear, step-by-step process:

  1. Identify the parameters: Determine the values of n, p, and x from the problem statement. In our case, n = 10, p = 0.45, and x = 8.
  2. Calculate the binomial coefficient: Use the formula (nx)=n!x!(n−x)!\binom{n}{x} = \frac{n!}{x!(n-x)!} to compute the number of ways to choose x successes from n trials. For our example, we calculated (108)=45\binom{10}{8} = 45.
  3. Calculate p^x: Raise the probability of success on a single trial (p) to the power of the desired number of successes (x). In our case, we calculated 0.45^8 ≈ 0.00016816920.
  4. Calculate (1-p)^(n-x): Raise the probability of failure on a single trial (1-p) to the power of the number of failures (n-x). For our example, we calculated 0.55^2 = 0.3025.
  5. Apply the formula: Multiply the results from steps 2, 3, and 4 to obtain the binomial probability P(x). In our case, we calculated P(8) = 45 * 0.00016816920 * 0.3025 ≈ 0.00229.

By following these steps, you can confidently calculate binomial probabilities for various scenarios.

The calculated probability of 0.00229 represents the likelihood of observing exactly 8 successes in 10 independent trials, given that the probability of success on each trial is 0.45. This probability is quite small, indicating that it is relatively unlikely to observe 8 successes in this experiment.

To gain a better understanding of this result, it's helpful to consider the context of the experiment. For instance, if this experiment represents a series of free throws in basketball, where a player has a 45% probability of making each shot, then the probability of making exactly 8 out of 10 free throws is only about 0.229%. This suggests that it would be a somewhat rare occurrence for this player to achieve such a high success rate in a series of 10 free throws.

Furthermore, we can compare this probability to the probabilities of other outcomes, such as getting 7 successes, 6 successes, or fewer. By calculating the binomial probabilities for these other outcomes, we can construct a probability distribution that provides a more complete picture of the likelihood of different numbers of successes. This distribution would reveal which outcomes are more likely and which are less likely, giving us a deeper understanding of the experiment's behavior.

The binomial probability distribution has a wide range of applications in various fields. Here are a few examples:

  • Quality Control: In manufacturing, binomial probability can be used to assess the quality of products. For example, if a factory produces light bulbs, the binomial distribution can help determine the probability of finding a certain number of defective bulbs in a batch.
  • Medical Research: In clinical trials, binomial probability is used to analyze the effectiveness of treatments. For instance, if a new drug is being tested, the binomial distribution can help determine the probability of a certain number of patients experiencing a positive outcome.
  • Marketing: Marketers use binomial probability to understand the success rates of advertising campaigns. For example, if a company sends out emails to potential customers, the binomial distribution can help determine the probability of a certain number of people clicking on the link.
  • Genetics: In genetics, binomial probability can be used to predict the inheritance of traits. For example, if two parents carry a gene for a particular trait, the binomial distribution can help determine the probability of their child inheriting the trait.
  • Polling and Surveys: When conducting polls or surveys, binomial probability can be used to estimate the margin of error. For instance, if a poll asks people whether they support a particular candidate, the binomial distribution can help determine the probability that the poll results accurately reflect the opinions of the population.

These are just a few examples of the many ways in which binomial probability is used in the real world. Its versatility and applicability make it a valuable tool for anyone working with data and making decisions under uncertainty.

When calculating binomial probabilities, it's crucial to avoid common mistakes that can lead to incorrect results. Here are some pitfalls to watch out for:

  • Misidentifying the parameters: Ensure you correctly identify the values of n, p, and x. A mistake in any of these values will lead to an incorrect probability calculation.
  • Incorrectly calculating the binomial coefficient: The binomial coefficient can be tricky to calculate manually, especially for large values of n and x. Double-check your calculations or use a calculator or software to compute the binomial coefficient accurately.
  • Using the wrong formula: The binomial probability formula applies specifically to situations with a fixed number of independent trials and two possible outcomes. If the conditions of the problem don't match these criteria, the binomial probability formula is not appropriate.
  • Rounding errors: Avoid rounding intermediate results during the calculation, as this can lead to significant errors in the final probability. Only round the final answer to the desired level of precision.
  • Misinterpreting the result: Remember that the binomial probability represents the likelihood of a specific outcome (exactly x successes). Don't confuse this with the probability of at least x successes or at most x successes, which require different calculations.

By being aware of these common mistakes, you can increase your accuracy and confidence when working with binomial probabilities.

In this comprehensive guide, we have explored the concept of binomial probability and learned how to calculate the probability of x successes in n independent trials of an experiment. We delved into the binomial probability formula, broke down the calculation into a step-by-step process, and discussed the importance of interpreting the results in context. Furthermore, we examined real-world applications of binomial probability and highlighted common mistakes to avoid.

By mastering the principles and techniques outlined in this guide, you are well-equipped to tackle a wide range of problems involving binomial probabilities. Whether you're analyzing quality control data, evaluating the effectiveness of a medical treatment, or predicting the outcome of an election, the knowledge you've gained will empower you to make informed decisions based on probability and statistics. Remember to practice these concepts with various examples to solidify your understanding and build your confidence in applying them to real-world scenarios.