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STATLABAcademyStatistics through data and R practice

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STATLAB Academy | Learn Statistics with R and Real-World Data

Lesson 16 of 16

Permutations, Combinations, and the Module 5 Capstone

Home/Modules/Module 5/Permutations, Combinations, and the Module 5 Capstone

Select the correct counting rule, then integrate probability, Bayes, reliability, and counting in a retail-risk decision.

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95 minutes100% path

Learning objectivesBy the end of this lesson, students will be able to:

1Define the events and probability quantities used in permutations, combinations, and the module 5 capstone.
2Select and calculate an appropriate probability rule with a visible denominator or assumption.
3Interpret the result for a business decision without overstating certainty or causation.

Theory

Core probability reasoning

Select the correct counting rule, then integrate probability, Bayes, reliability, and counting in a retail-risk decision.

Explanation 1

Permutation: Selection where order matters.

Explanation 2

Combination: Selection where order does not matter.

Explanation 3

Decision synthesis: Connect reproducible calculations to a cautious recommendation.

Terminology

Concept map

Permutation

Selection where order matters.

Assigning first, second, and third reviewers.

Combination

Selection where order does not matter.

Choosing a three-person review team.

Decision synthesis

Connect reproducible calculations to a cautious recommendation.

Digital retail risk capstone.

Notation and formulas

Permutation and combination

Key relationship

nPr = n!/(n-r)!; nCr = n!/[r!(n-r)!]

Define every event before substituting values, then validate the result and denominator.

Step 1Validate n >= r >= 0.
Step 2Decide whether order changes the outcome.
Step 3Calculate and interpret without assuming equal likelihood.

Interactive Mission

counting rule selector

Does order matter?Is replacement allowed?Do stages have equal choices?

Recommended rule

Combination

The recommendation follows whether order changes the outcome, selections can repeat, and stage choices are constant.

Interactive Mission

permutation combination comparator

Available items nSelected items r

Permutation nPr

336

Order matters

Combination nCr

56

Order does not matter

Inputs must satisfy n >= r >= 0 and be nonnegative integers.

Interactive Mission

probability risk capstone

Define events and denominatorsCalculate three empirical probabilitiesCalculate a complement and unionBuild a contingency tableEvaluate independence descriptivelyCalculate a Bayes posteriorEvaluate redundant reliabilityApply a counting ruleCreate two probability displaysWrite recommendation and limitations

Capstone progress

0/10

AI may coach formulas and debug R, but the learner owns the final report and recommendation.

Worked example

Digital retail risk

Scenario

A fictional retailer evaluates alerts, payment routes, backup reliability, and review staffing.

The recommendation must name assumptions, limitations, and operational trade-offs.

R connection

R note: Digital Retail Risk and Reliability Decision

Run and edit the code, inspect intermediate output, and use the movable R Code Guide while practising.

Live R Lab

Digital Retail Risk and Reliability Decision

Combine empirical probability, event rules, Bayes updating, reliability, and counting in one reproducible analysis.

Editable R code

# Module 5 R Note 13: Probability capstone
orders <- read.csv("module5_probability_capstone_synthetic.csv")
print(head(orders))
print(str(orders))
print(colSums(is.na(orders)))

alert <- orders$alert_triggered
issue <- orders$confirmed_issue
high_cart <- orders$cart_band == "200 or more"
print(data.frame(
  event = c("Alert", "Issue", "High cart", "Alert or high cart", "Issue given alert"),
  probability = c(mean(alert), mean(issue), mean(high_cart),
                  mean(alert | high_cart), sum(issue & alert) / sum(alert))
))

counts <- table(Alert = alert, Issue = issue)
print(addmargins(counts))
print(prop.table(counts, margin = 1))

component_reliability <- 0.985
backup_count <- 3
system_reliability <- 1 - (1 - component_reliability)^backup_count
print(data.frame(component_reliability, backup_count, system_reliability))
print(data.frame(review_teams = choose(8, 3), ordered_review_sequences = factorial(3)))

barplot(prop.table(table(orders$payment_method)),
        main = "Payment-method probabilities", ylab = "Proportion")
barplot(c(prior = mean(issue), posterior = sum(issue & alert) / sum(alert)),
        main = "Issue probability before and after alert", ylab = "Probability")
print("State event definitions, independence assumptions, limitations, and a decision-focused recommendation.")

Results

Ready to run

Ready to run

Ready to run. Results, errors, tables, plots, and fallback messages will appear here.

Define events.

Calculate and visualize probabilities.

State assumptions and limitations.

Dataset provenance

Original synthetic data generated for STATLAB Academy. No textbook data used.

Interpretation safeguard

Counting a large number of possibilities does not imply that all possibilities are equally likely.

Reflection

Pause and write

Where could an unclear event definition or assumption change the conclusion in permutations, combinations, and the module 5 capstone?

Name the event, denominator, probability interpretation, and one operational limitation.

Assistant

Ask your probability coach

Help me reason through Permutations, Combinations, and the Module 5 Capstone by asking questions about my event definitions, denominator, formula choice, assumptions, and interpretation. Do not give a quiz answer before I submit.

Exit check

Lesson checkpoint

Use a hint before submitting if needed. Full calculations and interpretations appear after submission.

Checkpoint

Module 5 Lesson 16 Checkpoint

Question 1 of 2. Answered 0/2. Passing score: 70%.

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Question 1: Numeric answer

mediumCalculate and responsibly interpret probability for a defined event.

How many ways can 3 reviewers be selected from 8 when order does not matter?

Resource

Recommended reading

Explore additional probability explanations and exercises from properly licensed or publisher resources.

Resource

Module 5 learning material

Download an original worksheet or guide for this lesson.

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