π GCSE Maths Β· Notes & worksheet
Probability.
Notes
Core rules
- P(not A) = 1 β P(A)
- Independent events: P(A and B) = P(A) Γ P(B)
- Mutually exclusive events: P(A or B) = P(A) + P(B)
- General: P(A or B) = P(A) + P(B) β P(A and B)
Tree diagrams
- Multiply along the branches, add between the branches that give the outcome you want.
- Without replacement: the second set of branches changes. After taking a red counter from 5 red and 3 blue, the second draw has 4 red and 3 blue out of 7.
- "At least one" is usually easiest as 1 β P(none).
Venn diagrams and set notation
| Symbol | Meaning |
|---|---|
| A β© B | A and B (the overlap) |
| A βͺ B | A or B (or both) |
| Aβ² | not A |
| ΞΎ | everything (the universal set) |
Fill the overlap first, then the "only" regions, then the outside.
Conditional probability
P(A | B) means "the probability of A given B has happened".
P(A | B) = P(A β© B) / P(B).
On a Venn diagram, the "given" event becomes your new total.
Independence test
A and B are independent if and only if P(A β© B) = P(A) Γ P(B).
Worked example
A bag contains 5 red and 3 blue counters. Two are taken without replacement. P(same colour) = (5/8)(4/7) + (3/8)(2/7) = 20/56 + 6/56 = 26/56 = 13/28.
Common mistakes
- Not changing the denominator on the second draw when there is no replacement.
- Adding along branches instead of multiplying.
- Using the whole total as the denominator for a conditional probability.
Exam tips
- Algebraic probability questions: form an equation, expand to a quadratic and solve. Reject any negative or non-integer solutions.
- Show the multiplications. Method marks are available even if the arithmetic slips.
Worksheet
Prints the questions only, without answers. Choose βSave as PDFβ in the print window to download.
Total: 24 marks. Give fractions in their simplest form unless stated.
- A bag has 5 red and 3 blue counters. Two counters are taken at random without replacement. Find the probability that both counters are the same colour. (3)
- Using the same bag, find the probability that at least one counter is blue. (2)
- A bag has 4 red and 6 green counters. Three counters are taken without replacement. Find the probability that all three are green. (2)
- 40 students were asked about French (F) and Spanish (S). 25 take French, 20 take Spanish and 8 take neither.
(a) Draw a Venn diagram and find how many take both. (3)
(b) A student who takes Spanish is chosen at random. Find the probability they also take French. (1) - P(A) = 0.5, P(B) = 0.4 and P(A β© B) = 0.15.
(a) Are A and B independent? Show your working. (2)
(b) Find P(A | B). (1) - 2% of a population have a condition. A test gives a positive result for 95% of people who have it, and for 4% of people who do not. Find the probability that a person chosen at random tests positive, and the probability that someone who tests positive actually has the condition (3 s.f.). (5)
- A bag contains 4 red counters and n blue counters. Two counters are taken at random without replacement. The probability that both are red is 2/15. Show that nΒ² + 7n β 78 = 0 and hence find n. (5)
Show mark scheme and answers (tutor copy)
Try the worksheet first, then check your work here.
- (5/8 Γ 4/7) + (3/8 Γ 2/7) (M1, M1) = 26/56 = 13/28. (A1)
- P(both red) = 20/56 (M1). P(at least one blue) = 1 β 20/56 = 36/56 = 9/14. (A1)
- 6/10 Γ 5/9 Γ 4/8 (M1) = 120/720 = 1/6. (A1)
- (a) Either or both: 40 β 8 = 32 (M1). Both = 25 + 20 β 32 = 13 (M1). Venn: French only 12, both 13, Spanish only 7, neither 8 (A1; total 40 β).
(b) 13/20. (B1) - (a) P(A) Γ P(B) = 0.2, which is not equal to 0.15 (M1), so A and B are not independent (A1).
(b) P(A | B) = 0.15/0.4 = 0.375. (B1) - P(positive) = 0.02 Γ 0.95 + 0.98 Γ 0.04 (M1, M1) = 0.019 + 0.0392 = 0.0582 (A1). P(condition | positive) = 0.019 / 0.0582 (M1) = 0.326 (A1). Many people who test positive do not have the condition because the condition is rare.
- P(both red) = 4/(n + 4) Γ 3/(n + 3) = 12/((n + 4)(n + 3)) (M1). 12/((n + 4)(n + 3)) = 2/15, so (n + 4)(n + 3) = 90 (M1). nΒ² + 7n + 12 = 90, so nΒ² + 7n β 78 = 0 (A1). (n + 13)(n β 6) = 0 (M1). n must be positive, so n = 6. (A1) Check: 4/10 Γ 3/9 = 2/15 β.