📊 AINMATH · Classroom Worksheet
Binomial Distribution
High school · Probability & Statistics
⏱ 20–25 min
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Learning goal — Understand what a binomial distribution is, be able to calculate the probability P(X=k)=ₙCₖpᵏ(1−p)ⁿ⁻ᵏ, and find the mean E(X)=np and variance V(X)=np(1−p).
When a trial with success probability p is repeated independently n times —
P(X=k) = ₙCₖ · pᵏ · (1−p)ⁿ⁻ᵏ E(X) = np V(X) = np(1−p)
💻 Access on a tablet → ainmath.com/binomial-distribution.html
1Calculating Binomial Probability
Q. When a coin is flipped 4 times, find the probability P(X=2) of getting exactly 2 heads. (p=0.5)
2Finding the Mean and Variance
① When n=20 and p=0.3, find E(X) and V(X).
② A certain product has a 10% defect rate. When inspecting 30 units, find the mean and variance of the number of defective items.
3Think Further
Q. A basketball player has an 80% free-throw success rate. If this player shoots 5 free throws, what's the average number of successes? Compare this to when p=0.5, and explain in one sentence how the shape of the distribution differs.
For teachers — cover before handing out to students
Answers & Solutions
Activity 1 — P(X=2) = ₄C₂×(0.5)²×(0.5)² = 6×0.25×0.25 = 0.375
Activity 2 — ① E(X)=20×0.3=6, V(X)=20×0.3×0.7=4.2 ② E(X)=30×0.1=3 items, V(X)=30×0.1×0.9=2.7
Activity 3 — E(X)=5×0.8=4 times. Since p=0.8 is greater than 0.5, the distribution isn't symmetric — it skews toward the high end (more successes).