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Intro Stats → Module 4 → Lesson 3

The Binomial Distribution

Learn to identify binomial experiments and calculate binomial probabilities

Lesson Objectives

By the end of this lesson, you will be able to:

1. What is a Binomial Experiment?

Definition: Binomial Experiment

A binomial experiment (or binomial setting) is a probability experiment that satisfies four specific conditions. When these conditions are met, the random variable X (number of successes) follows a binomial distribution.

Four Conditions for a Binomial Experiment

  1. Fixed Number of Trials (n): The experiment consists of a fixed number of identical trials
  2. Two Outcomes: Each trial has only two possible outcomes: "success" or "failure"
  3. Independent Trials: The trials are independent (outcome of one doesn't affect others)
  4. Constant Probability (p): The probability of success (p) is the same for each trial
Remember with "FTIC":
  • Fixed number of trials
  • Two outcomes (success/failure)
  • Independent trials
  • Constant probability

Example 1: Identifying Binomial Experiments

Scenario A: Flip a fair coin 20 times and count the number of heads.

  • Fixed n = 20 trials
  • Two outcomes: heads (success) or tails (failure)
  • Independent: each flip doesn't affect the next
  • Constant p = 0.5 for each flip
  • BINOMIAL!

Scenario B: Draw 5 cards from a deck WITHOUT replacement and count the number of aces.

  • Fixed n = 5 trials
  • Two outcomes: ace (success) or not ace (failure)
  • NOT independent: removing cards changes probabilities
  • NOT constant p: P(ace) changes after each draw
  • NOT BINOMIAL

Scenario C: A student guesses on 10 multiple-choice questions, each with 4 choices. Count correct answers.

  • Fixed n = 10 trials
  • Two outcomes: correct (success) or incorrect (failure)
  • Independent: answering one question doesn't affect others
  • Constant p = 0.25 for each question
  • BINOMIAL!

2. Binomial Probability Formula

If X follows a binomial distribution with n trials and success probability p, we can calculate the probability of exactly k successes using the binomial formula.

Binomial Probability Formula

P(X = k) = C(n, k) × p^k × (1-p)^(n-k)

Where:

  • n = number of trials
  • k = number of successes we want
  • p = probability of success on each trial
  • (1-p) = probability of failure on each trial (sometimes written as q)
  • C(n,k) = combination formula = n! / (k!(n-k)!)
Understanding the Formula:

The formula has three parts:

  • C(n,k): Number of ways to arrange k successes in n trials
  • p^k: Probability of k successes
  • (1-p)^(n-k): Probability of (n-k) failures

Example 2: Using the Binomial Formula

Problem: You flip a fair coin 5 times. What is the probability of getting exactly 3 heads?

Step 1: Verify binomial conditions

  • n = 5 (fixed trials)
  • Two outcomes (heads/tails)
  • Independent flips
  • p = 0.5 (constant)

Step 2: Identify values

  • n = 5, k = 3, p = 0.5

Step 3: Calculate C(5,3)

C(5,3) = 5! / (3!(5-3)!) = 5! / (3!2!) = (5×4×3×2×1) / ((3×2×1)(2×1)) = 120 / 12 = 10

Step 4: Apply the formula

P(X = 3) = C(5,3) × (0.5)³ × (0.5)²

= 10 × 0.125 × 0.25

= 10 × 0.03125

= 0.3125 or 31.25%

Interpretation: There's a 31.25% chance of getting exactly 3 heads in 5 coin flips.

Example 3: Multiple-Choice Test

Problem: A student guesses on a 10-question multiple-choice test. Each question has 4 choices. What's the probability of guessing exactly 4 questions correctly?

Setup: n = 10, k = 4, p = 0.25 (1/4 chance of guessing correctly)

Calculate C(10,4):

C(10,4) = 10! / (4!6!) = 210

Apply formula:

P(X = 4) = 210 × (0.25)⁴ × (0.75)⁶

= 210 × 0.00390625 × 0.17798

= 210 × 0.000695

≈ 0.146 or 14.6%

Interpretation: There's about a 14.6% chance of guessing exactly 4 out of 10 correctly.

3. Mean and Standard Deviation of Binomial Distribution

For binomial distributions, there are simple formulas to find the mean and standard deviation without using the general formulas from Lesson 2.

Binomial Mean (Expected Value)

μ = E(X) = np

Interpretation: On average, expect np successes in n trials

Binomial Standard Deviation

σ = √(np(1-p)) = √(npq)

Note: q = 1 - p is the probability of failure

Example 4: Mean and SD for Coin Flips

Scenario: Flip a fair coin 100 times. Find the mean and standard deviation of the number of heads.

Given: n = 100, p = 0.5

Mean:

μ = np = 100 × 0.5 = 50 heads

Standard Deviation:

σ = √(np(1-p)) = √(100 × 0.5 × 0.5) = √25 = 5 heads

Interpretation: If you flip a coin 100 times, you expect about 50 heads on average, with typical variation of about 5 heads. Most of the time, you'll get between 45 and 55 heads.

Example 5: Quality Control

Scenario: A factory produces light bulbs, 2% of which are defective. If we randomly select 500 bulbs, how many defects do we expect, and what's the standard deviation?

Given: n = 500, p = 0.02 (2% defect rate)

Mean:

μ = np = 500 × 0.02 = 10 defective bulbs

Standard Deviation:

σ = √(np(1-p)) = √(500 × 0.02 × 0.98) = √9.8 ≈ 3.13 defective bulbs

Interpretation: We expect about 10 defective bulbs on average, with typical variation of about 3 bulbs. Finding 7-13 defects would be fairly normal.

4. Common Binomial Scenarios

Typical Applications of Binomial Distribution:
  • Coin flips: Counting heads (or tails) in n flips
  • Multiple-choice tests: Counting correct guesses
  • Quality control: Counting defective items in a batch
  • Free throws: Counting made shots out of n attempts
  • Survey responses: Counting "yes" responses in n responses (if sampling with replacement)
  • Medical trials: Counting patients who improve out of n patients
When NOT to Use Binomial:
  • Sampling without replacement: Probabilities change (use hypergeometric instead)
  • More than two outcomes: Use multinomial distribution
  • Variable number of trials: Number of trials must be fixed
  • Non-constant probability: Success probability must stay the same

Check Your Understanding

Question 1: A basketball player makes 70% of free throws. She takes 20 shots. Is this binomial? If yes, what are n and p?

Answer: Yes, it's binomial.

  • Fixed n = 20 trials
  • Two outcomes: make or miss
  • Independent: each shot doesn't affect the next
  • Constant p = 0.70

n = 20, p = 0.70

Question 2: You roll a die until you get a 6. Is this binomial? Why or why not?

Answer: No, not binomial. The number of trials is NOT fixed—you keep rolling until you get a 6, which could be 1 roll, 10 rolls, or 100 rolls. This violates the "fixed n" condition. (This follows a geometric distribution instead.)

Question 3: For a binomial distribution with n = 50 and p = 0.4, what is μ?

Answer: μ = np = 50 × 0.4 = 20

On average, we expect 20 successes out of 50 trials when p = 0.4.

Question 4: If you flip a coin 16 times, what is the standard deviation of the number of heads?

Answer:

σ = √(np(1-p)) = √(16 × 0.5 × 0.5) = √4 = 2 heads

The number of heads typically varies by about 2 from the expected value of 8.

Question 5: Why is C(n,k) needed in the binomial formula?

Answer: C(n,k) counts the number of different ways to arrange k successes among n trials. For example, getting 3 heads in 5 flips can happen in C(5,3) = 10 different sequences (HHHTT, HHTHT, HTHHT, etc.). We multiply by this to account for all possible arrangements that give us exactly k successes.

Summary

Key Takeaways:
  • Binomial experiments have 4 conditions: Fixed n, Two outcomes, Independent, Constant p
  • Binomial probability formula: P(X = k) = C(n,k) × p^k × (1-p)^(n-k)
  • Mean of binomial: μ = np
  • Standard deviation: σ = √(np(1-p))
  • Common applications: coin flips, quality control, guessing on tests
  • Not binomial if: sampling without replacement, variable trials, or non-constant probability
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