Binomial Distribution
Number of successes in n independent trials.
Key idea
Number of successes in n independent trials.
In depth
The number of successes in n independent trials with the same success probability p: P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ. Mean np, variance np(1 − p).
Worked example
Five coin flips, P(exactly 3 heads) = C(5, 3)(1/2)⁵ = 10/32.
Formula
Real-world uses
- Estimating how many out of 20 patients respond to a treatment
- Quality control sampling
Common mistake
Using it when trials are not independent or p changes.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Binomial Distribution?
Number of successes in n independent trials. It is a topic at the advanced level in Probability & Statistics.
What is the formula for Binomial Distribution?
The key formula is P(X=k) = C(n,k) pᵏ(1−p)ⁿ⁻ᵏ.
Where is Binomial Distribution used in real life?
Real-world examples: Estimating how many out of 20 patients respond to a treatment; Quality control sampling.
What should I learn before Binomial Distribution?
Make sure you are comfortable with Bernoulli Distribution, the previous topic in Probability & Statistics.
What is a common mistake with Binomial Distribution?
Using it when trials are not independent or p changes.
How can I practise Binomial Distribution?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Binomial Distribution?
Move on to Geometric Distribution next.
Related advanced topics in Probability & Statistics
- Quartiles and IQR – Quartiles split data into four parts; IQR = Q3 − Q1.
- Box Plots – Five-number summary visualised.
- Outliers – Values beyond 1.5 IQR from the quartiles.
- Variance – Average squared deviation from the mean.
- Standard Deviation – Square root of variance; typical spread.
- Sample vs Population Variance – Divide by n−1 for sample variance.