Introduction to Binomial Distribution

What is a Binomial Distribution?

A binomial distribution is used when we count the number of successes in a fixed number of independent trials.
Each trial must have the same probability of success.

In general, a binomial distribution is written as:

X ~ B(n, p)

Where:

• n = number of trials
• p = probability of success
• q = 1 − p = probability of failure
• X = number of successes

Conditions for Binomial Distribution

A random variable follows a binomial distribution if ALL conditions are satisfied:

1. There are n independent trials
2. Each trial has only two outcomes (Success / Failure)
3. Probability of success p is constant
4. Random variable X counts number of successes

    

Most Common Exam Assumptions

Memorise this section! All questions follow the two most commonly tested assumptions which are:

1. Trials are carried out independently
2. The probability of success is constant

    

Discrete Random Variable vs Binomial Distribution

Discrete Random Variable

• Takes countable values
• Can take values like 0, 1, 2, 3…
• Does NOT always follow binomial distribution

Binomial Distribution

• A special type of discrete random variable
• Counts number of successes
• Must follow binomial conditions
• Written as X ~ B(n, p)

Important: All binomial distributions are discrete random variables,
but not all discrete random variables are binomial distributions.

How to Identify Binomial Distribution Questions

Ask yourself these 4 questions:

1. Is there a fixed number of trials?
2. Are there only two outcomes?
3. Is probability constant?
4. Are trials independent?

If all answers are YES → Use Binomial Distribution

Calculator Usage (Graphing Calculator)

To calculate binomial probabilities using calculator:

Calculator → 2nd → VARS → DISTR

• binompdf
• binomcdf

binompdf vs binomcdf

binompdf

Used to find probability of EXACTLY x successes

P(X = x)

Example:

Probability of exactly 3 successes

binompdf(n, p, 3)

binomcdf

Used to find probability of cumulative outcomes

P(X ≤ x)

Example:

Probability of at most 3 successes

binomcdf(n, p, 3)

When to Use Which (Exam Tips)

• Exactly → use binompdf
• At most → use binomcdf
• At least → use 1 − binomcdf
• More than → use 1 − binomcdf

Important Note

To calculate binomial distribution using the graphing calculator:

• Press 2nd
• Press VARS
• Select binomcdf or binompdf

Do take note that pdf and cdf are completely different functions.