Explanation 1
Point probability: Probability of exactly x successes.
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Lesson 9 of 18
Calculate point probabilities and connect n and p to center, spread, and skew.
Core lesson
Calculate point probabilities and connect n and p to center, spread, and skew.
Explanation 1
Point probability: Probability of exactly x successes.
Explanation 2
Mean: Expected success count np.
Explanation 3
Shape: How probability mass changes with n and p.
Terminology
Probability of exactly x successes.
Exactly five late services.
Expected success count np.
Expected late services.
How probability mass changes with n and p.
Right-skew for small p.
Notation and formulas
P(X=x)=C(n,x)p^x(1-p)^(n-x)
Define X, its support, parameters, units, and assumptions before substituting values.
Interactive Mission
Calculate a point probability for a fixed-trial 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.
Interactive Mission
Change n and p to inspect center, spread, and skew.
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.
Worked example
Scenario
A clinic models no-shows among 24 stable appointments.
Model assumptions matter more than a visually plausible curve.
R connection
Run and edit the original script, inspect intermediate output, and keep exact and approximate results clearly labelled.
Live R Lab
Translate exact, cumulative, interval, and upper-tail statements.
Ready to run
Dataset provenance
Original synthetic data generated for STATLAB Academy. No textbook data used.
Interpretation safeguard
Use p for the defined success event and keep n and x in the valid integer support.
Reflection
Name the random variable, support, parameters, units, dependence assumptions, and one limitation.
Assistant
Coach me through Binomial PMF, Mean, Variance, and Shape 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
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