How do you count all the ways to get k successes?
You will be able to: Calculate exact and cumulative binomial probabilities with correct boundaries.
How do you count all the ways to get k successes?
In three independent trials with p=0.20, exactly one success can occur first, second or third. Each ordering has the same probability.
A useful starting point: When is a count binomial? →
Words and symbols before equations
- Combination C(n,k)
- Number of ways to choose the k success positions among n trials.
- Factorial n!
- Product n×(n−1)×…×1, with 0!=1.
- Binomial probability
- C(n,k)pᵏ(1−p)ⁿ⁻ᵏ.
- Tail probability
- A sum over values at one end of the distribution.
What this picture assumes
A fixed number n of independent, identical trials, each with success probability p, is assumed. X counts successes. A cutoff beyond n is allowed: its probability follows the selected event, not a truncated count.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- Mean np = 1.2 successes; SD √[np(1−p)] = 0.849 successes. P(X = 2) = 0.288.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
One ordering with k successes and n−k failures has probability pᵏ(1−p)ⁿ⁻ᵏ. The combination factor counts all distinct success-position choices.
For n=3, p=0.20 and k=1, C(3,1)=3, so P(X=1)=3(0.20)(0.80)²=0.384. The coefficient must not be omitted.
To find at most k, add probabilities from 0 through k. For at least k, subtract the cumulative probability through k−1 from 1. A calculator’s binomial PDF gives one exact count; its CDF accumulates at-or-below probabilities.
A worked example, step by step
X is binomial with n=4 and p=0.5. Find P(X=2) and P(X≥3).
- C(4,2)=4!/(2!2!)=6.
- P(X=2)=6(0.5)²(0.5)²=6/16=0.375.
- P(X≥3)=P(X=3)+P(X=4)=(4+1)/16.
- The result is 5/16=0.3125, including both 3 and 4 successes.
The complement of X≥k is X≤k−1 for an integer count. Omitting k−1 loses the boundary mass.
Why multiply by C(n,k)?
Compare with an explanation
Because the successes can occupy different trial positions, each producing a distinct sequence.
Predict. Change one thing. Explain.
Switch among exactly, at most and at least for the same k. Check which bars are included and compare their summed probability.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Mean np = 1.2 successes; SD √[np(1−p)] = 0.849 successes. P(X = 2) = 0.288.
| k successes | P(X=k) | Included in request? |
|---|---|---|
| 0 | 0.216 | No |
| 1 | 0.432 | No |
| 2 | 0.288 | Yes |
| 3 | 0.064 | No |
A fixed number n of independent, identical trials, each with success probability p, is assumed. X counts successes. A cutoff beyond n is allowed: its probability follows the selected event, not a truncated count.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the probability values, reference groups, graph scales or model assumptions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionX is binomial with n=3, p=0.4. Find P(X=0), P(X=1) and P(X≥2).
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Compare with the answer and four-point rubric
- 1 point: P(X=0)=0.6³=0.216.
- 1 point: P(X=1)=3×0.4×0.6²=0.432.
- 1 point: P(X≥2)=1−0.216−0.432=0.352.
- 1 point: This includes exactly two and exactly three successes, assuming independent equal-p trials.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does the combination count?
Possible sets of success positions.
RECALL 2What does a binomial CDF calculate?
At-or-below probability.
RECALL 3Why shade selected bars?
To make the inequality and included boundaries explicit.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you count all the ways to get k successes?
- P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ.
- P(X≤k)=sum from 0 through k.
- P(X≥k)=1−P(X≤k−1).
Remember: The complement of X≥k is X≤k−1 for an integer count. Omitting k−1 loses the boundary mass.
Conditions: A fixed number n of independent, identical trials, each with success probability p, is assumed. X counts successes. A cutoff beyond n is allowed: its probability follows the selected event, not a truncated count.
Refresh Kid · AP Statistics Unit 2 · Objectives 2.10.C, 2.10.E · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 2.10, objectives 2.10.C, 2.10.E. Framework effective Fall 2026, checked September 17, 2026. Unit 2 includes probability, random variables, probability models and introductory sampling distributions; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Assumptions about independence, replacement and equal likelihood are stated before calculations. Simulation estimates fluctuate. Discrete probability is summed; continuous probability is area. Sampling distributions and randomization distributions use different repetition mechanisms. Formal inference comes in later units.
The Organic Chemistry Tutor companion title and destination were checked; the full video was not reviewed. Khan Academy’s destination was checked, but its lesson content was not fully readable by the research tool. OpenStax provides optional reference reading. No provider scripts, questions or graphics were copied. Refresh Kid is not affiliated with these providers.
GitHub’s 3D website collection informed optional spatial inspection. Our original three-toss outcome cube uses self-hosted Three.js with its MIT license. Eight corners represent eight equally likely sequences of three fair independent tosses. Conditioning removes ineligible sequences; camera rotation never changes probabilities. Quantitative graphs remain 2D to avoid perspective distortion. Complete labeled diagrams, outcome lists and explanations remain available without 3D.
Independent teacher review and observation of students remain pending. Technical checks do not certify statistical accuracy, accessibility or learning effectiveness. This is a review edition.
Released AP Statistics questions and scoring guides are optional. Older exams use the earlier framework, so check alignment before selecting parts. All practice on this page is original, not official AP material.
Learn → Explore → Practice → Review is informed by the IES learning guide. This implementation has not yet been evaluated with learners.
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