Binomial probabilities
Objective. Calculate point, cumulative and tail probabilities.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Binomial probabilities for X~B(n,p) can be found using the formula P(X=r)=ⁿCᵣpʳ(1−p)ⁿ⁻ʳ for a single value, or cumulative tables/calculator functions for P(X≤r) and tail probabilities like P(X>r)=1−P(X≤r). For example, with X~B(5,0.3), P(X=2)=⁵C₂(0.3)²(0.7)³. Cumulative probabilities are useful for ranges, since P(X≥r)=1−P(X≤r−1), avoiding the need to sum many individual terms by hand. Always identify whether the question wants exactly r, at most r, or at least r before calculating.
Worked example
X~B(6,0.4). Find the binomial probability P(X=2) exactly.
- 1Use P(X=r)=ⁿCᵣpʳ(1−p)ⁿ⁻ʳ
- 2P(X=2)=⁶C₂(0.4)²(0.6)⁴
- 3⁶C₂=15, so 15×0.16×0.1296
- 4P(X=2)=0.31104
Answer: 0.31104
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students compute P(X≥r) as 1−P(X≤r) instead of 1−P(X≤r−1), wrongly excluding the value r itself. Always adjust the boundary by one when switching from ≥ to the complement of ≤.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.