Question

For the given values of \(n\) and \(p\) computer plot graphs of the binomial density function for the probability of \(x\) successes in \(n\) Bernoulli trials with probability \(p\) of success. $$n=50, p=4 / 5$$

Short answer

Plot the graph using a tool like Python's 'matplotlib' with the binom.pmf function for n=50 and p=0.8.

Step-by-step solution

Understand the Binomial Distribution

The Binomial Distribution represents the probability of having exactly x successes in n independent Bernoulli trials with a success probability of p. The probability mass function (pmf) can be given by: \[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \] where \( \binom{n}{x} \) is the binomial coefficient.

Define the Parameters

For our problem, we are given \( n = 50 \) (the number of trials), and \( p = \frac{4}{5} \) (the probability of success). Therefore, \( p = 0.8 \) and the probability of failure is \( 1 - p = 0.2 \).

Calculate the Probabilities

To plot the binomial distribution, probabilities for each possible value of \( x \) (where \( x \) ranges from 0 to 50) must be calculated. Using the binomial pmf formula, the probability \( P(X = x) \) for each \( x \) can be calculated.

Use Software or a Graphing Tool

To plot the graph, software like Python, R, or even a graphing calculator can be utilized. In Python, for example, the 'scipy.stats' library can be used to compute these probabilities and plot the graph using 'matplotlib'.

Implement the Code

Here is a sample code snippet in Python:```pythonimport numpy as npimport matplotlib.pyplot as pltfrom scipy.stats import binomn = 50p = 0.8x = np.arange(0, n+1)probabilities = binom.pmf(x, n, p)plt.bar(x, probabilities)plt.xlabel('Number of Successes (x)')plt.ylabel('Probability P(X = x)')plt.title('Binomial Distribution (n=50, p=0.8)')plt.show()```This code will display a bar plot of the binomial distribution.

Key concepts

Bernoulli Trials

Bernoulli trials are a sequence of independent experiments where each experiment has two possible outcomes: success or failure. Each trial has the same success probability, denoted by \( p \), and, consequently, the failure probability is \( 1 - p \). One of the simplest examples of a Bernoulli trial is flipping a coin, where getting heads can be considered a success and tails a failure.
Bernoulli trials are the building blocks for the binomial distribution.
In the context of the provided exercise, we have 50 Bernoulli trials, each with a probability of success of 0.8.

Probability Mass Function

The probability mass function (pmf) of a binomial distribution gives the probability of observing exactly \( x \) successes in \( n \) Bernoulli trials. The pmf for the binomial distribution is given by the formula:
\[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \]
This formula calculates the likelihood of getting a specific number of successes in a fixed number of trials.
For example, in our problem where \( n = 50 \) and \( p = 0.8 \), we can use this formula to find the probabilities for each possible value of \( x \) (from 0 to 50).

• \( \binom{n}{x} \) is the binomial coefficient
• \( p^x \) is the probability of success raised to the power of \( x \)
• \( (1-p)^{n-x} \) is the probability of failure raised to the power of \( (n - x) \)

This function helps us build and understand the shape of the binomial distribution.

Binomial Coefficient

The binomial coefficient \( \binom{n}{x} \) represents the number of ways to choose \( x \) successes out of \( n \) trials. It is also known as a 'combination' and is mathematically given by:
\[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \]
where \( ! \) denotes factorial (the product of all positive integers up to that number).
In our problem:

• \( \binom{50}{0} = 1 \)
• \( \binom{50}{1} = 50 \)
• \( \binom{50}{25} = 1,264,322,234,167,310 \)

The binomial coefficient is significant because it tells us how many different ways we can achieve a specific number of successes in our set of trials.
It essentially weights the probability of each possible outcome by the number of ways that outcome can occur.

Scipy Library

The scipy library in Python is a powerful tool for scientific and technical computing. It includes modules for optimization, integration, interpolation, eigenvalue problems, algebraic equations, and many other tasks.
One of the modules, 'scipy.stats', offers a huge range of statistical distributions and functions, including the binomial distribution.
In our exercise, the `scipy.stats.binom` class is used to compute the probability mass function for the binomial distribution. Here’s how it works:

• Import necessary modules: `from scipy.stats import binom`
• Define the parameters for the binomial distribution: `n = 50`, `p = 0.8`
• Calculate probabilities for each possible number of successes: `probabilities = binom.pmf(x, n, p)`

Finally, `matplotlib` is used to create a bar plot of these probabilities, giving a visual representation of the binomial distribution.
The scipy library makes it straightforward to perform sophisticated statistical analyses and create visualizations with relatively simple code.