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. $$n=50, p=1 / 5$$

Short answer

Plot the distribution \( P(X = x) = \binom{50}{x} \left( \frac{1}{5} \right)^x \left( \frac{4}{5} \right)^{50-x} \) for \( x = 0 \) to \( 50 \).

Step-by-step solution

Define the binomial density function

The binomial density function or probability mass function (pmf) for a binomial random variable is given by: \[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \] where \( \binom{n}{x} \) is the binomial coefficient, \( p \) is the probability of success, and \( n \) is the number of trials.

Substitute given values

For the given values, substitute \( n = 50 \) and \( p = \frac{1}{5} \) into the binomial density function: \[ P(X = x) = \binom{50}{x} \left( \frac{1}{5} \right)^x \left( \frac{4}{5} \right)^{50-x} \]

Compute the binomial coefficient

The binomial coefficient \( \binom{50}{x} \) can be computed using the formula: \[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \] This coefficient represents the number of ways to choose \( x \) successes out of \( 50 \) trials.

Calculate probability for each x

Substitute values of \( x \) ranging from 0 to 50 into the formula \( P(X = x) = \binom{50}{x} \left( \frac{1}{5} \right)^x \left( \frac{4}{5} \right)^{50-x} \). This will give the probability of having \( x \) successes in 50 trials.

Plot the probability mass function

Use the computed probabilities to plot the binomial density function. Each \( x \) value along the horizontal axis represents the number of successes, and the corresponding \( P(X = x) \) values on the vertical axis represent the probability of those successes.

Key concepts

probability mass function

The probability mass function (PMF) is a function that gives the probability of a discrete random variable being exactly equal to some value. In the context of a binomial distribution, the PMF is used to calculate the probability of having exactly \( x \) successes in \( n \) independent Bernoulli trials, each with the same probability of success \( p \). For the binomial distribution, the PMF is defined as:

\[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \]
Where:

• \( n \): number of trials
• \( x \): number of successes
• \( p \): probability of success
• \( 1-p \): probability of failure

This formula combines the probability of successes, failures, and the various ways these can happen over multiple trials.

binomial coefficient

The binomial coefficient, represented as \( \binom{n}{x} \), is a key part of the binomial distribution formula. It calculates the number of ways to choose \( x \) successes out of \( n \) trials. Mathematically, it’s defined as:
\[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \]

The exclamation mark (\(!\)) denotes a factorial, which means multiplying all positive integers up to that number. For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)

The binomial coefficient helps in determining the different ways of arranging successes and failures in a given number of trials. Without it, we can't make meaningful calculations for binomial probabilities.

Bernoulli trials

A Bernoulli trial is a random experiment where there are only two possible outcomes: success or failure. Each trial is independent, meaning the outcome of one trial does not affect the outcome of another. Key points to remember about Bernoulli trials:

• The probability of success \( p \) remains the same for each trial.
• The trials are repeated a fixed number of times \( n \).
• Each trial results in either success or failure.

For example, when flipping a fair coin, the probability of landing heads (success) is 0.5, and the probability of landing tails (failure) is also 0.5. Conducting several coin flips can be considered a series of Bernoulli trials. Understanding Bernoulli trials is fundamental to grasping the concept of the binomial distribution, as a binomial distribution is simply the probability distribution of the number of successes in a given number of Bernoulli trials.

plotting probability

Once the probabilities are computed for all possible values of \( x \) (the number of successes), we can plot these probabilities to visualize the binomial distribution. Here’s how to do it:

• Identify the range of \( x \) values (from 0 to \( n \), where \( n \) is the number of trials).
• Calculate the probability for each \( x \) using the binomial formula.
• Use a graphing tool or software to plot \( x \) values on the horizontal axis.
• Plot the corresponding probabilities \( P(X = x) \) on the vertical axis.

You will often see a series of vertical bars or points indicating these probabilities. Each bar’s height corresponds to the probability of that specific number of successes.

This plot provides an intuitive understanding of how likely different outcomes are and lets you see the overall distribution of probabilities. Typically, for a large number of trials, the binomial distribution plot will resemble a bell-shaped curve, especially if \( p \) is around 0.5. This graphical representation makes interpreting and understanding the binomial distribution much easier.