Binomial Distribution Overview

Overview

This lesson explains how to use Desmos.com to graph a binomial distribution and calculate probabilities using a coin-toss example.

There is a fixed number of trials, denoted by ( n ).
Each trial has only two outcomes: success (probability ( p )) or failure (probability ( q )); ( p + q = 1 ).
Trials are independent and conducted under identical conditions.
The random variable ( x ) represents the number of successes in ( n ) trials.

Tossing a fair coin 50 times (( n = 50 )), with ( p = 0.5 ) for heads, ( q = 0.5 ) for tails.

Go to desmos.com and select the Graphing Calculator.
Enter the binomial distribution (select Distribution → Binomial).
Input number of trials (50) and probability of success (0.5).
Use the zoom fit button to adjust the graph window.
Click points on the graph to see probabilities for specific numbers of successes.

Probability of getting 30 or more heads (( x \geq 30 )): set min to 30, max to 50; result ≈ 0.101 (10.1%).
Probability of 28 or fewer heads (( x \leq 28 )): set min to 0, max to 28; result ≈ 0.839 (83.9%).
Probability of getting between 25 and 35 heads (( 25 \leq x \leq 35 )): set min to 25, max to 35; result ≈ 0.555 (55.5%).
Probability of exactly 30 heads (( x = 30 )): set min and max to 30; result ≈ 0.042 (4.2%).

Binomial Experiment — an experiment with a fixed number of independent, identical trials, each with two possible outcomes.
Trial — a single occurrence of the experiment.
Success — the outcome of interest in a trial.
Probability of Success (( p )) — chance of getting a success in one trial.
Probability of Failure (( q )) — chance of getting a failure in one trial (( q = 1 - p )).
Random Variable (( x )) — number of successes in ( n ) trials.

Practice using Desmos to graph and analyze other binomial distributions with different parameters.
Record and interpret probabilities for additional binomial scenarios.