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Binomial Probability

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Binomial Probability

Study Guide

Key Definition

Binomial probability refers to the probability of exactly $x$ successes on $n$ repeated trials in an experiment which has two possible outcomes. Here, $n C x = \frac{n!}{(n - x)!\,x!}$. If the probability of success on an individual trial is $p$, then the binomial probability is $n C x \cdot p^x \cdot (1 - p)^{n - x}$.

Important Notes

Binomial probability applies only when there are exactly two possible outcomes in a trial.
Each trial in a binomial experiment is independent of each other.
The experiment consists of a fixed number n of identical trials.
The probability p of success remains the same for every trial.

Mathematical Notation

$n C x$ indicates the number of different combinations of $x$ objects selected from a set of $n$ objects.

$n C x = \frac{n!}{(n - x)!\,x!}$

Remember to use proper notation when solving problems.

Why It Works

The binomial probability formula calculates the probability of obtaining exact successes in trials, by multiplying the probability of success, the probability of failure, and the number of ways the success can occur.

Remember

The formula for binomial probability is $n C x \cdot p^x \cdot (1 - p)^{n - x}$.

Quick Reference

Binomial Probability Formula:$n C x \cdot p^x \cdot (1 - p)^{n - x}$

Understanding Binomial Probability

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Video explanation of this concept

Beginner

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Beginner Explanation

Binomial probability is the probability of exactly $x$ successes in $n$ repeated trials of a binomial experiment.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the probability of getting 3 heads when you toss a coin 5 times?

Real-World Problem

Question Exercise

Intermediate

Candy Bag Scenario

A bag contains 5 red candies and 7 blue candies. If you randomly draw 3 candies with replacement, what is the probability that all 3 candies are blue?

Advanced Challenge

Thinking Exercise

Advanced

Think About This

In a multiple-choice test of 20 questions, each question has 4 options, and only one of them is correct. What is the probability of guessing exactly 5 correct answers?

Recap

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