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. $$n=50, p=4 / 5$$

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.

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.

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.

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.