P(A or B) = P(A) + P(B) − P(A and B). The subtraction is the whole rule, and a two-way table is where it becomes obvious: the corner cell sits in the row total AND in the column total, so adding the two counts those people twice.
Every question here is a survey of twenty to a hundred people, split two ways — walked or took the bus, on time or late. "What percent took the bus AND were late" is one cell. "What percent took the bus OR were late" is three, and a student who adds the row to the column without subtracting the corner gets a percent that can pass 100 — which is the arithmetic telling them what the rule is for.
Why this sheet works
The mixed setting puts both questions on the SAME table, one word apart. That is the arrangement where a wrong reading shows: on a page of unions only, a student who has read "or" as "and" still writes a number and gets no signal back.
How this one works
One question from a sheet like this one, worked a step at a time — with a sentence saying what just happened and why.
Use the addition rule. Write each answer as a whole percent.
1
Helmet
None
Total
Bikes
18
12
30
Scooters
9
21
30
Total
27
33
60
What percent are Bikes OR Helmet?
%
2
Won
Lost
Total
Team A
27
3
30
Team B
12
18
30
Total
39
21
60
What percent are Team A OR Won?
%
3
Renewed
Returned
Total
Fiction
28
12
40
Facts
16
24
40
Total
44
36
80
What percent are Fiction OR Renewed?
%
4
Chess
Cards
Total
Juniors
27
3
30
Seniors
12
18
30
Total
39
21
60
What percent are Juniors OR Chess?
%
The addition rule · The addition ruleAnswer key
1. What percent are Bikes OR Helmet? → 65% (30 + 27 − 18 = 39 of 60)2. What percent are Team A OR Won? → 70% (30 + 39 − 27 = 42 of 60)3. What percent are Fiction OR Renewed? → 70% (40 + 44 − 28 = 56 of 80)4. What percent are Juniors OR Chess? → 70% (30 + 39 − 27 = 42 of 60)
