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# Binomial Distribution

Probability and Statistics
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Probability Distributions
schedule Mar 10, 2022
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The binomial distribution is a discrete distribution which models the number of successes in a fixed number of independent trials. It can be described by two parameters:

$n$ = number of trials

$p$ = probability of success

The following conditions apply for the binomial distribution to be appropriate:

• For each trial there are only 2 outcomes: success or failure

• Each trial is independent (result of one trial does not affect the result of another)

• The number of trials is fixed: $n$

• Each trial has the same probability of success: $p$

# Formula

The binomial formula is represented as follows:

$$\mathbb{P}(X=x)= \tbinom{n}{x}p^xq^{n-x}$$

where:

$n$ = number of trials

$p$ = probability of success

$q$ = probability of failure = $1 - p$

# Example

Let us toss a coin four times. We define random variable $X$ as the number of heads obtained from the four tosses. Assume that this is a fair coin and the probability of getting a heads is 0.5 and tails is 0.5.

To find the probability of getting exactly 2 heads (i.e. $\mathbb{P}(X=2)$):

\begin{align*} \mathbb{P}(X=2) &= \tbinom{4}{2}0.5^20.5^{4-2}\\ &= 0.375 \end{align*}

By plotting the probability mass function, we can see the probability of getting the various numbers of heads in our four tosses of the coin:

As per our calculation from the formula, we can see that the probability of getting exactly 2 heads in our 4 tosses is 0.375.

The above graph can be plotted using Python's SciPy and Matplotlib libraries:

 # Import the relevant librariesfrom scipy.stats import binomimport matplotlib.pyplot as plt# Specify n (number of trials) and p (probability of success)n = 4p = 0.5# Specifying the ranger_vals = list(range(n + 1))# Calculate the pmf values and store in a listpmf = [binom.pmf(r, n, p) for r in r_vals ]# Plotting a bar chart with Matplotlibplt.bar(r_vals, pmf)# Adding the axis labelsplt.xlabel(r'$x$')plt.ylabel(r'$p_X(x)$')plt.show() 
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