Explanation 1
Fixed n: The number of trials is known in advance.
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Lesson 8 of 18
Diagnose the fixed-n, binary, constant-p, independence, and success-count assumptions.
Core lesson
Diagnose the fixed-n, binary, constant-p, independence, and success-count assumptions.
Explanation 1
Fixed n: The number of trials is known in advance.
Explanation 2
Constant p: Each trial uses the same success probability.
Explanation 3
Independence: One result does not change another's probability.
Terminology
The number of trials is known in advance.
Twenty scheduled services.
Each trial uses the same success probability.
Stable late rate.
One result does not change another's probability.
Separate service outcomes.
Notation and formulas
X ~ Binomial(n,p)
Define X, its support, parameters, units, and assumptions before substituting values.
Interactive Mission
Review fixed n, binary outcomes, constant p, independence, and success count.
P(X = x)
0.2182
F(x) = P(X <= x)
0.6296
E(X)
4
SD(X)
1.7889
| x | P(X=x) | Included in F(4) |
|---|---|---|
| 0 | 0.0115 | Yes |
| 1 | 0.0576 | Yes |
| 2 | 0.1369 | Yes |
| 3 | 0.2054 | Yes |
| 4 | 0.2182 | Yes |
| 5 | 0.1746 | No |
| 6 | 0.1091 | No |
| 7 | 0.0545 | No |
| 8 | 0.0222 | No |
| 9 | 0.0074 | No |
| 10 | 0.002 | No |
| 11 | 0.0005 | No |
| 12 | 0.0001 | No |
| 13 | 0 | No |
| 14 | 0 | No |
| 15 | 0 | No |
| 16 | 0 | No |
| 17 | 0 | No |
| 18 | 0 | No |
| 19 | 0 | No |
| 20 | 0 | No |
Interpretation: Custom PMF, discrete uniform, Bernoulli, or binomial: inspect support and trial structure before choosing.
Transformed mean: 18; transformed SD: 3.5777; two-variable variance with covariance: 22.4. Standard deviations were not added directly.
Worked example
Scenario
A manager counts late outcomes among 20 services.
A convenient count is not automatically binomial.
R connection
Run and edit the original script, inspect intermediate output, and keep exact and approximate results clearly labelled.
Live R Lab
Simulate Bernoulli trials and validate a binomial probability model.
Ready to run
Dataset provenance
Original synthetic data generated for STATLAB Academy. No textbook data used.
Interpretation safeguard
Do not choose binomial solely because the outcome is a count.
Reflection
Name the random variable, support, parameters, units, dependence assumptions, and one limitation.
Assistant
Coach me through Recognizing a Binomial Model by asking about X, support, parameters, assumptions, probability notation, exact versus approximate status, and interpretation. Do not provide a quiz answer before submission or write my final capstone recommendation.
Exit check
Use conceptual hints before submitting. Full formulas, calculations, and interpretation appear afterward.
Checkpoint
Question 1 of 2. Answered 0/2. Passing score: 70%.
Resource
Explore additional discrete-distribution explanations from properly licensed or publisher resources.
DownloadResource
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