13 marks
The events A and B are mutually exclusive, with P(A) = 0.4 and P(B) = 0.35
Find
- P(A ∩ B)
- P(A ∪ B)
- P(A′ ∩ B′)
Show answer
a) 0 b) 0.75 c) 0.25
Probability
Venn diagrams, tree diagrams, two-way tables, mutually exclusive and independent events.
12 original questions, 68 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.
The questions
24 marks
The events A and B are independent, with P(A) = 0.3 and P(B) = 0.6
Find
- P(A ∩ B)
- P(A ∪ B)
- P(A ∩ B′)
Show answer
a) 0.18 b) 0.72 c) 0.12
34 marks
A security system has three alarms that work independently. Each alarm fails to sound with probability 0.05
An intruder sets off the system. Find the probability that
- at least one of the alarms fails to sound,
- exactly one of the alarms fails to sound.
Show answer
a) 0.143 (3 s.f.) b) 0.135 (3 s.f.)
45 marks
A sports club has 50 members. 23 members play tennis, 18 play squash and 7 play both.
- Draw a Venn diagram to represent this information.
A member is chosen at random. Find the probability that the member plays
- exactly one of these sports,
- neither sport.
Show answer
a) tennis only 16, both 7, squash only 11, neither 16 b) 27/50 c) 8/25
55 marks
A bag contains 5 red counters and 3 blue counters. Two counters are taken from the bag at random, one after the other, without replacement.
- Complete the tree diagram.
- Find the probability that the two counters are the same colour.
- Find the probability that at least one of the counters is blue.
Show answer
a) first: red 5/8, blue 3/8; after red: red 4/7, blue 3/7; after blue: red 5/7, blue 2/7 b) 13/28 c) 9/14
6
6 marks
The table shows how 120 sixth-form students usually travel to school.
| Year 12 | Year 13 | |
|---|---|---|
| Walk | 18 | 22 |
| Bus | 30 | 26 |
| Car | 12 | 12 |
A student is chosen at random. Find the probability that the student
- travels by bus,
- is in Year 13 and travels by car,
- walks or is in Year 12, or both.
- Determine whether the events “the student travels by car” and “the student is in Year 12” are independent.
Show answer
a) 7/15 b) 1/10 c) 41/60 d) P(car) × P(Year 12) = 0.2 × 0.5 = 0.1 = P(car and Year 12), so they are independent
76 marks
On any school day the probability that it rains is 0.3. If it rains, the probability that Zara’s bus is late is 0.4. If it does not rain, the probability that her bus is late is 0.15
- Complete the tree diagram to represent this information.
- Find the probability that Zara’s bus is late on a randomly chosen school day.
- Two school days are chosen at random. Assuming that the days are independent, find the probability that her bus is late on exactly one of the two days.
Show answer
a) rain 0.3, no rain 0.7; after rain: late 0.4, on time 0.6; after no rain: late 0.15, on time 0.85 b) 0.225 c) 0.349 (3 s.f.)
85 marks
The Venn diagram shows the probabilities for two events A and B, where p and q are probabilities.
Given that P(A) = 0.4
- find the value of p and the value of q,
- determine whether A and B are independent,
- find P(A ∪ B′).
Show answer
a) p = 0.15, q = 0.3 b) P(A) × P(B) = 0.4 × 0.45 = 0.18, which is not 0.15 = P(A ∩ B), so not independent c) 0.7
97 marks
The events A, B and C have P(A) = 0.5, P(B) = 0.4 and P(C) = 0.2
The events A and B are independent. The events A and C are mutually exclusive. The events B and C are independent.
- Find P(A ∩ B).
- Draw a Venn diagram to show the events A, B and C, giving the probability for each region.
- Find P([B ∪ C]′).
Show answer
a) 0.2 b) A only 0.3, A ∩ B 0.2, B only 0.12, B ∩ C 0.08, C only 0.12, outside 0.18 (A and C do not overlap) c) 0.48
107 marks
Box 1 contains 4 green counters and 6 white counters. Box 2 contains only green and white counters. One counter is taken at random from each box.
The probability that the two counters are the same colour is 0.52
- Find the probability of taking a green counter from Box 2.
- Complete the tree diagram.
- Find the probability that at least one of the counters is green.
The event G is that the counter from Box 1 is green. The event S is that the two counters are the same colour.
- Determine whether G and S are independent.
Show answer
a) 0.4 b) each Box 2 pair of branches: green 0.4, white 0.6 c) 0.64 d) P(G ∩ S) = 0.16 but P(G) × P(S) = 0.4 × 0.52 = 0.208, so not independent
118 marks
A college surveys 100 A-level students about three subjects: Maths (M), Physics (P) and Chemistry (C).
- 52 study Maths, 27 study Physics and 36 study Chemistry
- 20 study Maths and Physics, 18 study Maths and Chemistry, and 11 study Physics and Chemistry
- 8 study all three
- Draw a Venn diagram to represent this information.
A student is chosen at random. Find the probability that the student studies
- exactly two of these subjects,
- none of these subjects.
- Determine whether studying Physics and studying Chemistry are independent.
Show answer
a) all three 8; M and P only 12; M and C only 10; P and C only 3; M only 22; P only 4; C only 15; none 26 b) 0.25 c) 0.26 d) P(P) × P(C) = 0.27 × 0.36 = 0.0972, but P(P ∩ C) = 0.11, so not independent
128 marks
A factory has two machines, X and Y. In any week, the events “X breaks down” and “Y breaks down” are independent. Machine X is more likely to break down than machine Y.
In a week, the probability that both machines break down is 0.12 and the probability that at least one machine breaks down is 0.58
- Find the probability that machine X breaks down in a week.
- Find the probability that exactly one machine breaks down in a week.
- Assuming that different weeks are independent, find the probability that neither machine breaks down in two weeks running.
Show answer
a) 0.4 b) 0.46 c) 0.176 (3 s.f.)
Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.