Exam-style questionsConditional Probability questions with mark schemes
Easiest first. Try each question on paper, then open its mark scheme: M marks are for method, A marks for accurate answers that follow from the method, B marks for independent results.
Question 1easy5 marksYear 13
The members of a leisure club were asked which activity they use most. The results are shown by age group.
| Swim | Weights | Classes |
|---|
| Under 40 | 48 | 72 | 40 |
|---|
| 40 and over | 60 | 30 | 50 |
|---|
A member is chosen at random. \(S\) is the event that the member mainly swims, \(W\) that the member mainly uses weights and \(U\) that the member is under 40.
(a) Find \(P(U' \mid S)\). (2)
(b) Find \(P(W \mid U)\). (1)
(c) Determine whether the events \(W\) and \(U\) are independent. (2)
Mark scheme and answer
(a) \(\dfrac{60}{108}\) M1
\(= \tfrac{5}{9}\) A1
(b) \(\dfrac{72}{160} = \tfrac{9}{20}\) B1
(c) \(P(W) = \dfrac{102}{300} = \tfrac{17}{50}\) M1
\(P(W\mid U) = \tfrac{9}{20} \ne \tfrac{17}{50}\) so not independent A1
(or \(P(W\cap U)=\tfrac{72}{300}=0.24\) vs \(P(W)P(U)=\tfrac{17}{50}\times\tfrac{8}{15}=0.181\), not equal: M1 A1)
Answer: (a) 5/9; (b) 9/20; (c) not independent
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Question 2easy5 marksYear 13
For events \(A\) and \(B\), \(P(A) = 0.4\), \(P(B) = 0.5\) and \(P(A \cup B) = 0.7\).
(a) Find \(P(A \cap B)\). (2)
(b) Find \(P(A \mid B)\). (2)
(c) State, with a reason, whether \(A\) and \(B\) are independent. (1)
Mark scheme and answer
(a) \(0.4 + 0.5 - 0.7 = 0.2\) M1A1
(b) \(\dfrac{0.2}{0.5} = 0.4\) M1A1
(c) \(P(A\mid B) = 0.4 = P(A)\), so independent B1
Answer: (a) 0.2; (b) 0.4; (c) independent
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Question 3easy5 marksYear 13
For events \(A\) and \(B\), \(P(A) = 0.45\), \(P(B) = 0.3\) and \(P(A \cap B) = 0.12\).
(a) Find \(P(A' \cap B)\). (1)
(b) Find \(P((A \cup B)')\). (2)
(c) Find \(P(B \mid A')\). (2)
Mark scheme and answer
(a) \(0.3 - 0.12 = 0.18\) B1
(b) \(1 - (0.45 + 0.3 - 0.12) = 0.37\) M1A1
(c) \(\dfrac{0.18}{0.55} = \tfrac{18}{55}\) M1A1
Answer: (a) 0.18; (b) 0.37; (c) 18/55
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Question 4medium6 marksYear 13
A scanner at a delivery depot is used to identify damaged parcels. 3% of parcels are damaged. A damaged parcel is flagged by the scanner with probability 0.92. A parcel that is not damaged is flagged with probability 0.06.
(a) Find the probability that a randomly chosen parcel is flagged. (3)
(b) Given that a parcel is flagged, find the probability that it is damaged. (2)
(c) Comment on your answer to (b) in the context of the depot rejecting every flagged parcel. (1)
Mark scheme and answer
(a) \(0.03\times0.92 + 0.97\times0.06\) M1A1
\(= 0.0858\) A1
(b) \(\dfrac{0.03\times0.92}{0.0858}\) M1
\(= 0.322\) (\(=\tfrac{46}{143}\)) A1
(c) About two-thirds of flagged parcels are not damaged (because damage is rare), so rejecting every flagged parcel would waste many good parcels; flagged parcels should be checked by hand B1
Answer: (a) 0.0858; (b) 0.322 (= 46/143); (c) most flagged parcels are undamaged, so check before rejecting
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Question 5medium5 marksYear 13
A bag contains 5 red counters and 7 blue counters. Two counters are taken at random, one after the other, without replacement.
(a) Find the probability that the second counter is red. (3)
(b) Given that the second counter is red, find the probability that the first counter was red. (2)
Mark scheme and answer
(a) \(\tfrac{5}{12}\times\tfrac{4}{11} + \tfrac{7}{12}\times\tfrac{5}{11}\) M1A1
\(= \tfrac{5}{12}\) A1
(b) \(\dfrac{\tfrac{20}{132}}{\tfrac{5}{12}}\) M1
\(= \tfrac{4}{11}\) A1
Answer: (a) 5/12; (b) 4/11
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Question 6medium5 marksYear 13
When serving, a tennis player gets her first serve in with probability 0.62. If the first serve is in, she wins the point with probability 0.75. If the first serve is out, she takes a second serve, which is in with probability 0.88; if it is in she wins the point with probability 0.55, and if it is out she loses the point.
(a) Find the probability that she wins a point on her serve. (3)
(b) Given that she wins the point, find the probability that her first serve was in. (2)
Mark scheme and answer
(a) \(0.62\times0.75 + 0.38\times0.88\times0.55\) M1A1
\(= 0.64892\) A1
(b) \(\dfrac{0.465}{0.64892}\) M1
\(= 0.717\) A1
Answer: (a) 0.64892; (b) 0.717
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Question 7medium4 marksYear 13
35% of the customers of an online bookshop are subscribers. The probability that a subscriber leaves a review is 0.4; the probability that a non-subscriber leaves a review is 0.15.
(a) Find the probability that a randomly chosen customer leaves a review. (2)
(b) A review is chosen at random. Find the probability that it was written by a subscriber. (2)
Mark scheme and answer
(a) \(0.35\times0.4 + 0.65\times0.15\) M1
\(= 0.2375\) A1
(b) \(\dfrac{0.14}{0.2375}\) M1
\(= 0.589\) A1
Answer: (a) 0.2375; (b) 0.589 (= 56/95)
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Question 8medium6 marksYear 13
80 students were asked which of French (\(F\)), Spanish (\(S\)) and German (\(G\)) they study. 3 study all three; 8 study French and Spanish only; 5 study French and German only; 6 study Spanish and German only; 14 study French only; 20 study Spanish only; 9 study German only; the rest study none of these.
(a) Draw a Venn diagram to represent this information. (2)
(b) A student is chosen at random. Find \(P(F \mid S')\). (2)
(c) Find the probability that the student studies exactly one of these languages, given that they study at least one. (2)
Mark scheme and answer
(a) Three intersecting circles with 3 in the centre, 8, 5, 6 in the pairwise regions B1
14, 20, 9 in the single regions and 15 outside B1
(b) \(P(S') = \dfrac{43}{80}\) M1
\(\dfrac{14+5}{43} = \tfrac{19}{43}\) A1
(c) \(\dfrac{14+20+9}{80-15} = \tfrac{43}{65}\) M1A1
Answer: (a) Venn with 3; 8, 5, 6; 14, 20, 9; 15; (b) 19/43; (c) 43/65
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Question 9hard7 marksYear 13
Events \(A\) and \(B\) satisfy \(P(A \cap B) = 0.12\), \(P(A \cap B') = x\), \(P(A' \cap B) = y\) and \(P(A' \cap B') = 0.42\). The events \(A\) and \(B\) are independent and \(P(A) \gt P(B)\).
(a) Find the value of \(x\) and the value of \(y\). (5)
(b) Find \(P(A \mid A \cup B)\). (2)
Mark scheme and answer
(a) \(x + y = 1 - 0.12 - 0.42 = 0.46\) B1
\((x+0.12)(y+0.12) = 0.12\) M1
Substitute \(y = 0.46 - x\): \(x^2 - 0.46x + 0.0504 = 0\) dM1
\(x = 0.28,\ y = 0.18\) (since \(P(A) \gt P(B)\)) A1A1
(b) \(\dfrac{P(A)}{P(A\cup B)} = \dfrac{0.40}{0.58}\) M1
\(= \tfrac{20}{29}\) A1
Answer: (a) x = 0.28, y = 0.18; (b) 20/29
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Question 10hard4 marksYear 12
A survey of adults recorded where they live and whether they own a car.
| Owns a car | Does not own a car |
|---|
| Town | \(3k\) | 60 |
|---|
| Village | 90 | 20 |
|---|
An adult is chosen at random. The events "lives in a town" and "owns a car" are independent.
(a) Find the value of \(k\). (3)
(b) Find the probability that a randomly chosen adult owns a car. (1)
Mark scheme and answer
(a) Independent \(\Rightarrow P(\text{car}\mid\text{town}) = P(\text{car}\mid\text{village})\) M1
\(\dfrac{3k}{3k+60} = \dfrac{90}{110}\) A1
\(330k = 270k + 5400\), \(k = 90\) A1
Alternative (AS method): \(P(T\cap C)=P(T)P(C)\): \(\dfrac{3k}{3k+170}=\dfrac{(3k+60)(3k+90)}{(3k+170)^2}\) M1, \(3k(3k+170)=(3k+60)(3k+90)\) A1, \(k=90\) A1
(b) \(\dfrac{270+90}{440} = \tfrac{9}{11}\) B1
Answer: (a) k = 90; (b) 9/11
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted. Independence only (no conditional probability is required), so this is AS/Year 1 content; the P(A∩B) = P(A)P(B) method is the expected AS route.
Question 11hard6 marksYear 13
On any day, the probability that it rains in the morning is 0.3. If it rains, the probability that Ben's bus is late is 0.25; if it does not rain, the probability that his bus is late is 0.1. Days are independent.
(a) Find the probability that Ben's bus is late on a given day. (2)
(b) Given that Ben's bus is late, find the probability that it rained that morning. (2)
(c) Find the probability that, on two consecutive days, the bus is late on exactly one day and it rained on that day. (2)
Mark scheme and answer
(a) \(0.3\times0.25 + 0.7\times0.1 = 0.145\) M1A1
(b) \(\dfrac{0.075}{0.145} = \tfrac{15}{29}\) M1A1
(c) P(rain and late on a day) = 0.075; P(not late) = 0.855 M1
\(2\times0.075\times0.855 = 0.12825\) A1
Answer: (a) 0.145; (b) 15/29; (c) 0.12825
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Question 12hard5 marksYear 12
The events \(A\) and \(B\) are independent.
(a) Prove that \(A\) and \(B'\) are also independent. (3)
(b) Given also that \(P(A) = 0.35\) and \(P(A\cup B) = 0.61\), find \(P(B)\). (2)
Mark scheme and answer
(a) \(P(A\cap B') = P(A) - P(A\cap B)\) M1
\(= P(A) - P(A)P(B)\) (independence) M1
\(= P(A)(1 - P(B)) = P(A)P(B')\), hence independent A1
(b) \(0.61 = 0.35 + P(B) - 0.35P(B)\) M1
\(0.65P(B) = 0.26\), \(P(B) = 0.4\) A1
Answer: (a) proof; (b) P(B) = 0.4
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted. Uses only the independence multiplication rule and the addition rule, so this is AS/Year 1 content.
Question 13hard5 marksYear 13
Two fair six-sided dice are rolled.
(a) Given that at least one die shows a 6, find the probability that the total is at least 9. (3)
(b) Given that the total is at least 9, find the probability that at least one die shows a 6. (2)
Mark scheme and answer
(a) Outcomes with at least one 6: 11 B1
Of these, total \(\ge 9\): (6,3),(6,4),(6,5),(6,6),(3,6),(4,6),(5,6) = 7 M1
\(P = \tfrac{7}{11}\) A1
(b) Total \(\ge9\): 10 outcomes M1
\(\tfrac{7}{10}\) A1
Answer: (a) 7/11; (b) 7/10
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Question 14challenging7 marksYear 13
Three events \(A\), \(B\) and \(C\) are such that \(A\) and \(C\) are mutually exclusive, and \(B\) and \(C\) are independent. It is given that \(P(A) = 0.3\), \(P(B) = 0.4\), \(P(A \cap B) = 0.1\), \(P(C \cap A' \cap B') = 0.15\) and \(P(A' \cap B' \cap C') = p\).
(a) Find \(P(B \cap C)\). (3)
(b) Find the value of \(p\). (2)
(c) Find \(P(B \mid A' )\). (2)
Mark scheme and answer
(a) Let \(P(B\cap C) = c\); \(P(C) = c + 0.15\) M1
Independence: \(c = 0.4(c + 0.15)\) M1
\(c = 0.1\) A1
(b) \(P(A\cup B\cup C) = 0.3 + 0.4 - 0.1 + 0.15 = 0.75\) M1
\(p = 0.25\) A1
(c) \(P(B\cap A') = 0.4 - 0.1 = 0.3\) M1
\(\dfrac{0.3}{0.7} = \tfrac37\) A1
Answer: (a) 0.1; (b) p = 0.25; (c) 3/7
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Question 15challenging7 marksYear 13
4% of the items made in a factory are defective. Every item passes through two independent inspections. At each inspection a good item is wrongly rejected with probability 0.02. A defective item is rejected at the first inspection with probability 0.85 and, if it passes the first, at the second inspection with probability 0.7. Items rejected at either inspection are removed.
(a) Find the probability that an item passes both inspections. (4)
(b) Given that an item passes both inspections, find the probability that it is defective. (2)
(c) Comment on the effectiveness of the inspection process. (1)
Mark scheme and answer
(a) Defective passes: \(0.04\times0.15\times0.3 = 0.0018\) M1A1
Good passes: \(0.96\times0.98^2 = 0.921984\) M1
Total \(= 0.923784\) A1
(b) \(\dfrac{0.0018}{0.923784}\) M1
\(= 0.00195\) A1
(c) Only about 0.19% of items sent out are defective, compared with 4% before inspection, so the process is effective (though some good items are lost) B1
Answer: (a) 0.923784; (b) 0.00195; (c) defective rate greatly reduced
Board notes: Set notation (\(A\cap B\), \(A\cup B\), \(A'\)) is used by all three boards; AQA sometimes writes \(\overline{A}\) for the complement. Exact fractions or decimals are accepted.
Original questions in the style of Edexcel 9MA0, independently checked before publishing. Not past-paper questions. Covers: Two-way tables, Conditional probability formula, Set notation, Tree diagrams and Bayes-type problems, Tree diagrams without replacement, Tree diagrams, Venn diagrams with three events, Venn diagrams and independence, Two-way tables and independence, Independence proof, Conditional probability with sample spaces, Multi-stage tree diagrams.
A Level Math Revision