11-15-2025, 04:44 PM
Chapter 15 — The Binomial Distribution
The Binomial Distribution is a way of modelling:
“How likely is it to get X successes out of N tries?”
Anytime an event repeats and each outcome is:
• success / failure
• yes / no
• hit / miss
• pass / fail
• heads / tails
— the binomial distribution applies.
It is one of the most important probability models in all of science.
---
15.1 When to Use the Binomial Distribution
You can use the binomial model when all 4 conditions apply:
1. There is a fixed number of trials (N).
2. Each trial has only two outcomes.
3. The probability of success stays constant (p).
4. All trials are independent.
Examples:
• flipping a coin 10 times
• rolling a die and counting sixes
• shooting 20 basketball free-throws
• checking whether seeds germinate
• genetic inheritance (dominant/recessive)
---
15.2 The Binomial Formula
In simple form:
P(X = k) = C(n, k) × p^k × (1−p)^(n−k)
Where:
• n = number of trials
• k = number of successes
• p = probability of success
• C(n, k) = number of ways to arrange k successes
You do NOT need full mastery yet — we focus on intuition.
---
15.3 Intuitive Explanation
Suppose a basketball player scores a free throw with probability 0.7.
If they take 5 shots, what’s the probability they score:
• exactly 3?
• at least 4?
• none?
These questions CANNOT be solved with simple multiplication.
There are multiple different paths.
The binomial model counts ALL possible paths correctly.
---
15.4 Example 1 — Coin Flips
Flip a fair coin 4 times.
What is P(exactly 2 heads)?
Here:
n = 4
k = 2
p = 1/2
P(X = 2) = C(4,2) × (1/2)^2 × (1/2)^2
C(4,2) = 6
So:
= 6 × 1/16
= 6/16
= 3/8
Meaning:
If you flip a coin 4 times repeatedly,
about 37.5% of all sets will contain exactly 2 heads.
---
15.5 Example 2 — Rolling Dice
Roll a die 10 times.
What’s the probability of getting exactly 3 sixes?
Here:
n = 10
k = 3
p = 1/6
P(X = 3) = C(10,3) × (1/6)^3 × (5/6)^7
You could leave the answer in this form.
That is fully acceptable for GCSE-level.
---
15.6 Example 3 — Genetics
A plant has a 0.25 probability of having a certain trait.
If 6 seeds are grown, what is the probability that exactly 2 show the trait?
n = 6
k = 2
p = 0.25
P(X=2) = C(6,2) × (0.25)^2 × (0.75)^4
Again, you can leave it like this or evaluate with a calculator.
---
15.7 Cumulative Probabilities
Often, questions ask:
• “At least 3 successes”
• “No more than 2 successes”
• “4 or more”
You sum up multiple binomial probabilities:
Example:
P(X ≥ 3) = P(3) + P(4) + P(5) + …
You can write answers as a sum — no need to compute every value unless requested.
---
15.8 Mean and Variance of a Binomial Distribution
These formulas are VERY useful:
Mean = np
Variance = np(1−p)
Standard deviation = √(np(1−p))
Example:
Coin flipped 100 times
n = 100, p = 0.5
Mean = 100×0.5 = 50
SD = √(100×0.5×0.5) = √25 = 5
Meaning:
Average heads = 50
Typical range = 45–55
---
15.9 When the Binomial Model Breaks
Do NOT use the binomial distribution if:
• probability changes each trial
• order matters in a dependent way
• trials are not independent
• sampling is without replacement from a small population
Example:
• Picking cards without replacement from a small deck → NOT binomial
• Choosing sweets from a bowl without replacement → NOT binomial
• Most games of chance → binomial
---
15.10 Exam-Style Questions
1. A coin is flipped 7 times.
Find P(exactly 4 heads).
2. An archer hits the target 70% of the time.
If they shoot 5 arrows:
(a) Find P(exactly 3 hits)
(b) Find P(at least 4 hits)
3. A die is rolled 12 times.
Find P(no sixes).
4. A plant has a 0.2 probability of mutation per seed.
If 8 seeds are grown, find P(exactly 1 mutation).
5. A quiz has 10 true/false questions.
Guessing randomly:
Find P(score ≥ 7 correct).
---
15.11 Chapter Summary
• Binomial distribution models repeated success/failure events
• Conditions: fixed trials, two outcomes, constant probability, independent trials
• Formula uses combinations to count arrangements
• Mean = np and variance = np(1−p)
• Used across science, finance, genetics, and exams
• Essential foundation for later probability models
You now understand one of the core tools of real-world probability.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
The Binomial Distribution is a way of modelling:
“How likely is it to get X successes out of N tries?”
Anytime an event repeats and each outcome is:
• success / failure
• yes / no
• hit / miss
• pass / fail
• heads / tails
— the binomial distribution applies.
It is one of the most important probability models in all of science.
---
15.1 When to Use the Binomial Distribution
You can use the binomial model when all 4 conditions apply:
1. There is a fixed number of trials (N).
2. Each trial has only two outcomes.
3. The probability of success stays constant (p).
4. All trials are independent.
Examples:
• flipping a coin 10 times
• rolling a die and counting sixes
• shooting 20 basketball free-throws
• checking whether seeds germinate
• genetic inheritance (dominant/recessive)
---
15.2 The Binomial Formula
In simple form:
P(X = k) = C(n, k) × p^k × (1−p)^(n−k)
Where:
• n = number of trials
• k = number of successes
• p = probability of success
• C(n, k) = number of ways to arrange k successes
You do NOT need full mastery yet — we focus on intuition.
---
15.3 Intuitive Explanation
Suppose a basketball player scores a free throw with probability 0.7.
If they take 5 shots, what’s the probability they score:
• exactly 3?
• at least 4?
• none?
These questions CANNOT be solved with simple multiplication.
There are multiple different paths.
The binomial model counts ALL possible paths correctly.
---
15.4 Example 1 — Coin Flips
Flip a fair coin 4 times.
What is P(exactly 2 heads)?
Here:
n = 4
k = 2
p = 1/2
P(X = 2) = C(4,2) × (1/2)^2 × (1/2)^2
C(4,2) = 6
So:
= 6 × 1/16
= 6/16
= 3/8
Meaning:
If you flip a coin 4 times repeatedly,
about 37.5% of all sets will contain exactly 2 heads.
---
15.5 Example 2 — Rolling Dice
Roll a die 10 times.
What’s the probability of getting exactly 3 sixes?
Here:
n = 10
k = 3
p = 1/6
P(X = 3) = C(10,3) × (1/6)^3 × (5/6)^7
You could leave the answer in this form.
That is fully acceptable for GCSE-level.
---
15.6 Example 3 — Genetics
A plant has a 0.25 probability of having a certain trait.
If 6 seeds are grown, what is the probability that exactly 2 show the trait?
n = 6
k = 2
p = 0.25
P(X=2) = C(6,2) × (0.25)^2 × (0.75)^4
Again, you can leave it like this or evaluate with a calculator.
---
15.7 Cumulative Probabilities
Often, questions ask:
• “At least 3 successes”
• “No more than 2 successes”
• “4 or more”
You sum up multiple binomial probabilities:
Example:
P(X ≥ 3) = P(3) + P(4) + P(5) + …
You can write answers as a sum — no need to compute every value unless requested.
---
15.8 Mean and Variance of a Binomial Distribution
These formulas are VERY useful:
Mean = np
Variance = np(1−p)
Standard deviation = √(np(1−p))
Example:
Coin flipped 100 times
n = 100, p = 0.5
Mean = 100×0.5 = 50
SD = √(100×0.5×0.5) = √25 = 5
Meaning:
Average heads = 50
Typical range = 45–55
---
15.9 When the Binomial Model Breaks
Do NOT use the binomial distribution if:
• probability changes each trial
• order matters in a dependent way
• trials are not independent
• sampling is without replacement from a small population
Example:
• Picking cards without replacement from a small deck → NOT binomial
• Choosing sweets from a bowl without replacement → NOT binomial
• Most games of chance → binomial
---
15.10 Exam-Style Questions
1. A coin is flipped 7 times.
Find P(exactly 4 heads).
2. An archer hits the target 70% of the time.
If they shoot 5 arrows:
(a) Find P(exactly 3 hits)
(b) Find P(at least 4 hits)
3. A die is rolled 12 times.
Find P(no sixes).
4. A plant has a 0.2 probability of mutation per seed.
If 8 seeds are grown, find P(exactly 1 mutation).
5. A quiz has 10 true/false questions.
Guessing randomly:
Find P(score ≥ 7 correct).
---
15.11 Chapter Summary
• Binomial distribution models repeated success/failure events
• Conditions: fixed trials, two outcomes, constant probability, independent trials
• Formula uses combinations to count arrangements
• Mean = np and variance = np(1−p)
• Used across science, finance, genetics, and exams
• Essential foundation for later probability models
You now understand one of the core tools of real-world probability.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive