(i) A Random Variable X Follows Binomial Distribution With 12 Trails. (a) Find The Value Of P Given That

(i) A Random Variable X Follows Binomial Distribution With 12 Trails. (a) Find The Value Of P Given That

When dealing with binomial distributions, one of the most common problems involves determining the probability of success, denoted as p, given certain information about the random variable X. In this context, suppose that a random variable X follows a binomial distribution with 12 trials, and we are tasked with finding the value of p given specific conditions such as the probability of a certain number of successes or the expected value. Understanding how to approach this problem is essential for students and professionals working in statistics, data analysis, and related fields.

---

Understanding the Binomial Distribution

What Is a Binomial Distribution?

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is characterized by two parameters:

• The number of trials, denoted as n (in this case, 12).
• The probability of success in a single trial, denoted as p.

The probability mass function (PMF) for a binomial distribution is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where k is the number of successes.

Key Properties of Binomial Distribution

• The expected value (mean): \( \mu = n p \)
• Variance: \( \sigma^2 = n p (1 - p) \)
• The distribution is discrete and symmetric when p = 0.5.

---

Problem Statement: Finding p Given Certain Conditions

Suppose that:

• The random variable \( X \sim \text{Binomial}(n=12, p) \)


• We are given some information about X (for example, the probability of a specific number of successes or the expected number of successes)


• Our goal is to determine the value of p based on this information

Example Scenario:
For instance, if it is given that the probability of getting exactly 3 successes is a certain value, or that the expected number of successes is known, we can set up an equation to solve for p.

---

Methods to Find the Value of p

Using the Probability of a Specific Number of Successes

Suppose you know: \[ P(X = k) = p_k \] for some k and p_k (a specific probability). You can then use the binomial PMF: \[ p_k = \binom{12}{k} p^k (1-p)^{12 - k} \] and solve for p.

Example:
If \( P(X=3) = 0.22 \), then:
\[ 0.22 = \binom{12}{3} p^3 (1-p)^9 \]
which simplifies to:
\[ 0.22 = 220 \times p^3 (1-p)^9 \]
and solving for p involves numerical methods or iterative solutions such as the Newton-Raphson method.

---

Using the Expected Value of X

The expected value of X for a binomial distribution is: \[ E[X] = n p \] If you are given the average number of successes, say \( \bar{x} \), then: \[ p = \frac{\bar{x}}{n} \] This approach is straightforward and often used when the mean is provided.

Example:
If the average number of successes in repeated experiments is 4.8:
\[ p = \frac{4.8}{12} = 0.4 \]

---

Using the Cumulative Probability

In some instances, you might be given the probability of X being less than or equal to a certain value: \[ P(X \leq k) = p_{c} \] You can then use binomial tables or statistical software to find p that satisfies this cumulative probability.

---

Practical Steps to Find p in the Given Scenario

Step 1: Identify the Given Data

• Is the probability of a specific number of successes provided?
• Is the expected number of successes given?
• Is the cumulative probability provided?

Step 2: Choose the Appropriate Method

• Use the probability mass function if a specific success probability is given.
• Use the mean if the average number of successes is provided.
• Use cumulative probabilities when applicable.

Step 3: Set Up the Equation

• For specific probabilities: \[ p_k = \binom{n}{k} p^k (1-p)^{n-k} \]
• For mean: \[ p = \frac{\text{mean}}{n} \]
• For cumulative probabilities: Use statistical software or binomial tables to find p.

Step 4: Solve for p

• Algebraic manipulation for simple cases.
• Numerical methods for complex equations.

---

Example Problem and Solution

Problem:
Suppose that in 12 trials, the probability of getting exactly 4 successes is 0.15. Find the value of p.

Solution:
Given:
\[ P(X=4) = 0.15 \]
Using the binomial PMF:
\[ 0.15 = \binom{12}{4} p^4 (1-p)^8 \]
Calculate \(\binom{12}{4}\):
\[ \binom{12}{4} = 495 \]
So:
\[ 0.15 = 495 \times p^4 (1-p)^8 \]
Divide both sides by 495:
\[ \frac{0.15}{495} \approx 0.000303 \]
Then:
\[ p^4 (1-p)^8 = 0.000303 \]
This equation cannot be solved algebraically easily; instead, use numerical methods or software (like calculator, Excel, R, or Python) to approximate p.

Using software:

• Set up the equation and iterate over possible p values to find the one that satisfies the equation.


• For instance, in Python, one can write a small script to find p numerically.

---

Conclusion

Determining the value of p in a binomial distribution when given specific information about the distribution involves understanding the key properties and applying the appropriate mathematical tools. Whether using the probability mass function, the expected value, or cumulative probabilities, the approach depends on the data provided. In real-world applications, statistical software greatly simplifies the process, especially for equations that lack closed-form solutions.

Summary of key points:

• Identify the given data: probability of successes, mean, or cumulative probability.




• Choose the appropriate method based on the data.




• Set up the binomial probability equation.




• Solve for p using algebraic manipulation or numerical methods.

By mastering these steps, students and practitioners can efficiently find the unknown probability p for binomial distributions with 12 trials or any other number of trials, enhancing their understanding of probabilistic models and their applications in various fields.