Customer Overlaps

Estimated reading time: 2 minutes

A business worksheet can contain every pairwise channel overlap yet still leave its unique-customer total unresolved. This fictional example shows the missing information.

Hold six reported counts fixed

Use 300 distinct people. 100 responded through A, 100 through B and 100 through C. Each pair—A/B, A/C and B/C—shares 50 people. These are inclusive pair counts: someone in all three channels belongs to every pair.

Let t be the unknown number responding through all three. The following mutually exclusive groups reproduce the six supplied counts:

Group	People
A only	t
B only	t
C only	t
A/B only	50 − t
A/C only	50 − t
B/C only	50 − t
All three	t
None	150 − t

For example, A’s total is t + (50 − t) + (50 − t) + t = 100. The inclusive A/B overlap is (50 − t) + t = 50. The other channels and pairs follow identically; all eight groups sum to 300.

Establish the exact missing range

Counts cannot be negative: the triple group requires t ≥ 0, while each pair-only group requires t ≤ 50. Conversely, every integer from 0 through 50 makes every displayed count nonnegative and preserves all six reports. Thus these are attainable alternatives, not merely loose limits.

The unique responders total 300 − (150 − t) = 150 + t. At t = 0 there are 150 unique responders; at t = 50 there are 200.

Carry that uncertainty into one expense test

Now stipulate that every unique responder buys exactly once, contributes $10 after variable costs, and that $1,700 is the only remaining expense. These are deterministic teaching assumptions, not conversion estimates or a complete borrowing budget.

The remaining amount is $10 × (150 + t) − $1,700 = −$200 + $10t. It is negative for integer t from 0 through 19, zero at 20, and positive from 21 through 50. The two endpoint worlds leave −$200 and +$300 respectively.

Ask for the missing intersection

Oscar Levin’s counting text explains inclusion–exclusion: add individual set sizes, subtract pairwise intersections, then add the triple intersection. The pair counts include triple members; they are not counts of people in exactly two channels.