Explanation 1
At most: X<=x.
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Lesson 10 of 18
Translate verbal probability statements into precise inclusive integer boundaries.
Core lesson
Translate verbal probability statements into precise inclusive integer boundaries.
Explanation 1
At most: X<=x.
Explanation 2
More than: X>x.
Explanation 3
Inclusive interval: a<=X<=b.
Terminology
X<=x.
No more than five late services.
X>x.
Six or more when x=5.
a<=X<=b.
Three through seven.
Notation and formulas
P(X>=x)=1-P(X<=x-1)
Define X, its support, parameters, units, and assumptions before substituting values.
Interactive Mission
Keep integer boundaries visible when calculating tails.
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
Extra staff are called when at least six services are late.
P(X>5) and P(X>=5) are different.
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
Always display the probability statement beside the answer so the boundary is auditable.
Reflection
Name the random variable, support, parameters, units, dependence assumptions, and one limitation.
Assistant
Coach me through Binomial Cumulative, Interval, and Tail Probabilities 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.
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