Grade 12 Probability Questions: Venn Diagrams, Trees and Counting

The rules

• Addition rule: P(A or B) = P(A) + P(B) − P(A and B).
• Mutually exclusive: P(A and B) = 0, so P(A or B) = P(A) + P(B).
• Independent: P(A and B) = P(A) × P(B).
• Complement: P(not A) = 1 − P(A).

Worked example: testing independence

P(A) = 0.4, P(B) = 0.5 and P(A and B) = 0.2. Since 0.4 × 0.5 = 0.2, the events are independent. Then P(A or B) = 0.4 + 0.5 − 0.2 = 0.7.

Always show the multiplication test before stating your conclusion — the test is the mark.

Worked example: counting

How many arrangements of the letters in EDUCATE are there? There are 7 letters with the letter E repeated twice, so 7!/2! = 2 520 arrangements.

For a 4-digit code from digits 0–9 with no repeats, use 10 × 9 × 8 × 7 = 5 040. If repeats are allowed, use 10⁴ = 10 000.

Exam tips

• Fill in a Venn diagram from the intersection outwards.
• In tree diagrams, check that each set of branches adds to 1.
• 'At least one' is usually easiest as 1 minus the probability of none.
• Say whether repetition is allowed before you start counting.