Apriori Algorithm: A Complete Solved Numerical Example

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Scenario: Market Basket Analysis

The Objective: Discover which book categories are frequently purchased together by ruthlessly filtering out unpopular combinations before generating recommendation rules.

Core Mechanics

▼

Support is the Entry Ticket: To survive, an itemset must appear frequently enough to pass your Minimum Support threshold. If it falls short, cross it off immediately—it is permanently discarded.
The Pruning Rule (Apriori Property): If a small itemset is infrequent, every larger set containing it is also infrequent. If $\{A, B\}$ fails, you never waste time scanning the data for $\{A, B, C\}$ !
Level-by-Level Growth: You must build itemsets strictly one step at a time. Frequent singles combine to form pairs; surviving pairs form triplets. You only ever build size $k+1$ using the winners from size $k$ .
Confidence Makes Rules: Finding popular itemsets is only half the job. To create an actual recommendation rule ( $A \Rightarrow B$ ), you filter by Confidence: "Given they already have A, how likely are they to also have B?"

Step 1: The Transaction Data

The Apriori algorithm requires a list of transactions to mine frequent patterns. In this scenario, we need a Minimum Support of 50% and a Minimum Confidence of 70%.

Data Point	Items (Comma-Separated)
P1	A, B, C
P2	A, C, D
P3	B, C, E
P4	A, B, C, D
P5	A, C, E
P6	B, D, E

Legend:A = AI & MLB = BusinessC = CodingD = Data ScienceE = Economics

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Step 2: Frequent Itemset Generation

We iteratively generate Candidate Itemsets ( $C_k$ ) and filter them by the absolute minimum support count (3) to find Frequent Itemsets ( $L_k$ ).

Iteration 1: Finding 1-Itemsets

1. Generate Candidates (

C_{1}

)Count

{A}	4
{B}	4
{C}	5
{D}	3
{E}	3

2. Filter by Min Support (

L_{1}

)≥ 3

{A}	4
{B}	4
{C}	5
{D}	3
{E}	3

Iteration 2: Finding 2-Itemsets

1. Generate Candidates (

C_{2}

)Count

{A, B}	2Drop
{A, C}	4
{A, D}	2Drop
{A, E}	1Drop
{B, C}	3
{B, D}	2Drop
{B, E}	2Drop
{C, D}	2Drop
{C, E}	2Drop
{D, E}	1Drop

2. Filter by Min Support (

L_{2}

)≥ 3

{A, C}	4
{B, C}	3

Step 3: Association Rules Generation

For every frequent itemset of size 2 or more, we generate all possible rules ( $X \rightarrow Y$ ) and calculate their confidence. If the confidence is $\ge 70\%,$ the rule is kept.

Rule ( $X \rightarrow Y$ )	Sup ( $X \cup Y$ )	Sup ( $X$ )	Confidence	Status
{A}→{C}	4	4	100 %	Keep
{C}→{A}	4	5	80 %	Keep
{B}→{C}	3	4	75 %	Keep
{C}→{B}	3	5	60 %	Drop

Final Takeaway

Look at the final two rows in Step 3 to see the ultimate Apriori exam trap: rule asymmetry! Even though {B} → {C} passes the 70% confidence threshold and is kept, the exact reverse rule {C} → {B} scores only 60% and gets dropped, proving mathematically that association rules are strictly one-way.