Determining The Value Of X For Greatest Relative Frequency In Binomial Distribution

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Understanding binomial distributions is crucial in probability and statistics. In this article, we will dive into the heart of binomial distributions, specifically focusing on identifying the value of x that yields the highest relative frequency in a discrete random variable. Let's consider a scenario where we have a discrete random variable X following a binomial distribution, denoted as X ~ B(4, 1/10). This notation tells us that we are dealing with four trials, each having a probability of success of 1/10. The core question we aim to answer is: Which value of x will result in the greatest relative frequency? To solve this, we will meticulously examine the probability mass function (PMF) of the binomial distribution, calculate probabilities for different values of x, and ultimately pinpoint the value that maximizes the relative frequency.

Understanding Binomial Distribution

A binomial distribution models the probability of obtaining a certain number of successes in a fixed number of independent trials. Each trial has only two possible outcomes: success or failure. In our case, X ~ B(4, 1/10), the parameters are:

  • n = 4 (number of trials)
  • p = 1/10 (probability of success in each trial)

The probability mass function (PMF) for a binomial distribution is given by:

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

Where:

  • P(X = x) is the probability of observing exactly x successes.
  • (n choose x) is the binomial coefficient, representing the number of ways to choose x successes from n trials, also written as n! / (x!(n - x)!).
  • p^x is the probability of x successes.
  • (1 - p)^(n - x) is the probability of n - x failures.

To find the value of x that maximizes the relative frequency, we need to compute P(X = x) for all possible values of x (0, 1, 2, 3, and 4) and compare the results. This involves plugging in the values of n and p into the formula and calculating the probabilities. We will look for the highest probability, which corresponds to the value of x with the greatest relative frequency.

Calculating Probabilities for Each Value of x

To identify the value of x that yields the highest relative frequency, we need to calculate the probabilities for each possible outcome. Given the binomial distribution X ~ B(4, 1/10), we have n = 4 trials and a success probability of p = 1/10. We will compute P(X = x) for x = 0, 1, 2, 3, and 4 using the formula:

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

H2: Probability of x = 0

For x = 0, we calculate the probability of having no successes in four trials:

P(X = 0) = (4 choose 0) * (1/10)^0 * (9/10)^4

(4 choose 0) = 4! / (0! * 4!) = 1

P(X = 0) = 1 * 1 * (9/10)^4 = (9/10)^4 = 0.6561

H2: Probability of x = 1

For x = 1, we calculate the probability of having exactly one success in four trials:

P(X = 1) = (4 choose 1) * (1/10)^1 * (9/10)^3

(4 choose 1) = 4! / (1! * 3!) = 4

P(X = 1) = 4 * (1/10) * (9/10)^3 = 4 * (1/10) * (729/1000) = 0.2916

H2: Probability of x = 2

For x = 2, we calculate the probability of having exactly two successes in four trials:

P(X = 2) = (4 choose 2) * (1/10)^2 * (9/10)^2

(4 choose 2) = 4! / (2! * 2!) = 6

P(X = 2) = 6 * (1/10)^2 * (9/10)^2 = 6 * (1/100) * (81/100) = 0.0486

H2: Probability of x = 3

For x = 3, we calculate the probability of having exactly three successes in four trials:

P(X = 3) = (4 choose 3) * (1/10)^3 * (9/10)^1

(4 choose 3) = 4! / (3! * 1!) = 4

P(X = 3) = 4 * (1/10)^3 * (9/10) = 4 * (1/1000) * (9/10) = 0.0036

H2: Probability of x = 4

For x = 4, we calculate the probability of having all four successes in four trials:

P(X = 4) = (4 choose 4) * (1/10)^4 * (9/10)^0

(4 choose 4) = 4! / (4! * 0!) = 1

P(X = 4) = 1 * (1/10)^4 * 1 = (1/10000) = 0.0001

Identifying the Greatest Relative Frequency

Now that we have calculated the probabilities for each value of x, let's summarize our findings:

  • P(X = 0) = 0.6561
  • P(X = 1) = 0.2916
  • P(X = 2) = 0.0486
  • P(X = 3) = 0.0036
  • P(X = 4) = 0.0001

By comparing these probabilities, it is evident that the highest probability occurs when x = 0. This means that the greatest relative frequency is observed when there are no successes in the four trials. Therefore, the value of x that will result in the greatest relative frequency is x = 0. This outcome aligns with our expectations, as a low probability of success (p = 1/10) suggests that observing zero successes is the most likely event.

Conclusion

In summary, for a discrete random variable X ~ B(4, 1/10), the value of x that will result in the greatest relative frequency is x = 0. This conclusion is derived from calculating and comparing the probabilities for all possible values of x using the binomial probability mass function. Understanding how to determine the most likely outcome in a binomial distribution is fundamental in various statistical applications, enabling us to make informed decisions based on probability.

In this context, the correct answer to the question is:

  • D. x = 0

This analysis showcases the importance of grasping the nuances of binomial distributions and their applications in real-world scenarios. By systematically calculating and comparing probabilities, we can effectively identify the most probable outcomes and gain valuable insights into the behavior of random variables.