Lesson Guide
Explanation
Thirty students were asked about two sports. Eighteen play soccer, twelve play tennis, and five play both. Start in the middle: the five who play both go in the overlap. The soccer circle must hold eighteen in total, so soccer only is eighteen minus five, which is thirteen. Tennis only is twelve minus five, which is seven. Inside the circles there are thirteen plus five plus seven, which is twenty five students, so five play neither, and they go outside the circles. Notice that eighteen plus twelve minus five also gives twenty five.
Your Turn
In a class of 28, 15 own a bike, 10 own a scooter and 4 own both. How many own neither? Check: 15 + 10 − 4 = 21, so 7.
Watch Out
18 + 12 = 30 counts the 5 who play both twice. Subtract the overlap once: 25, not 30.
Explanation
Now pick one of the thirty students at random. Soccer and tennis means the student plays both, which is the overlap: five out of thirty, or one sixth. Soccer or tennis means the student plays at least one of them, so it is everything inside the circles: twenty five out of thirty, or five sixths. In maths, or always includes both. As a rule, the probability of S or T is the probability of S plus the probability of T, minus the probability of S and T.
Your Turn
From the same diagram, find P(soccer). Check: 18/30 = 3/5.
Watch Out
"Or" includes the overlap. 13/30 + 7/30 = 20/30 leaves out the 5 who play both.
Explanation
Not soccer means everything outside the soccer circle. That is the seven who play tennis only, and the five outside both circles: twelve out of thirty, or two fifths. Do not forget the outside region. Tennis only means the crescent of the tennis circle that is not in the soccer circle: seven out of thirty. Compare that with tennis, the whole circle, which is twelve out of thirty.
Your Turn
Find P(not T). Check: 13 + 5 = 18, so 18/30 = 3/5.
Watch Out
Read for the word "only". Tennis is 12; tennis only is 7.
Explanation
A two-way table holds exactly the same information. The rows are soccer and not soccer. The columns are tennis and not tennis. The overlap, five, goes in the soccer and tennis cell. Soccer only, thirteen, goes in soccer and not tennis. Tennis only, seven, goes in not soccer and tennis. The five outside both go in not soccer and not tennis. Then add along the rows and down the columns. The totals, eighteen and twelve, are the whole circles, and the grand total is thirty.
Your Turn
From the table, find P(not S and not T). Check: 5/30 = 1/6.
Watch Out
The row total 18 is the whole soccer circle. The "S, not T" cell is 13, the soccer-only crescent.
Explanation
Going the other way, read the four inside cells of the table. Dog and cat, six, goes in the overlap. Dog and no cat, fourteen, is dog only. Cat and no dog, ten, is cat only. No dog and no cat, twenty, goes outside. Now any probability can be read from either picture. A dog or a cat is fourteen plus six plus ten, which is thirty out of fifty, or three fifths. You can check it with one minus the probability of neither: one minus twenty fiftieths is also three fifths. Quiz and exercises now.
Your Turn
Find P(cat but no dog). Check: 10/50 = 1/5.
Watch Out
Put the "and" cell in the overlap first. Writing the row total 20 as "dog only" counts the 6 twice.
📌 Key Points — this lesson in a nutshell
- Fill a Venn diagram from the middle out: the overlap first, then each "only" region, then the outside.
- n(A or B) = n(A) + n(B) − n(A and B). Adding the circles counts the overlap twice.
- And = the overlap. Or = everything inside the circles, overlap included. Not A = everything outside circle A, including the region outside both. Only = a crescent, not a whole circle.
- A two-way table holds the same four numbers: A and B, A and not B, B and not A, neither. Its row and column totals are whole circles.
- Probability = (number in the region) ÷ (number in the whole group).
Step 1: Introduction
Step 1 of 5
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