Conditional Probability - Worksheets, Questions and Revision

16 original exam-style questions - 14 pages of questions with a full mark scheme - free printable PDF.

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H15 Conditional Probability

EDEXCEL 1ST0 · Calculator allowed · about 90 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
A gym surveyed 80 members on whether they attend on weekday mornings and whether they attend on weekend mornings. The two-way table shows the results.
Weekend morningsNo weekend morningsTotal
Weekday mornings241640
No weekday mornings103040
Total344680
(a)Write down the number of members who attend on both weekday mornings and weekend mornings.(1)
(b)One member is chosen at random from those who attend weekend mornings. Find the probability that this member also attends weekday mornings.(2)
(c)Explain what is meant by P(weekday mornings | weekend mornings) in the context of this table.(1)
(Total for Question 1 is 4 marks)
2
A drawer contains 7 socks: 4 black and 3 grey, all otherwise identical. Nadia takes a sock at random from the drawer and does not put it back. She then takes a second sock at random from the drawer.
(a)Find the probability that both socks are black.(2)
(b)Find the probability that both socks are grey.(2)
(c)Find the probability that Nadia takes one sock of each colour.(2)
(Total for Question 2 is 6 marks)
3
A box contains 9 pens: 5 blue and 4 black. Tomasz takes a pen at random from the box and does not replace it. He then takes a second pen at random from the box. The diagram shows the blank structure of a tree diagram for this situation.
1st pick 2nd pick
(a)Complete the tree diagram: label each of the six branches with its outcome (Blue or Black) and write its probability.(3)
(b)Use your tree diagram to find the probability that Tomasz takes two pens of the same colour.(3)
(Total for Question 3 is 6 marks)
4
150 customers at a cafe were asked whether they bought a hot drink and whether they bought a pastry. The two-way table shows the results.
PastryNo pastryTotal
Hot drink543690
No hot drink184260
Total7278150
(a)Write down P(Pastry).(1)
(b)Find P(Pastry | Hot drink).(2)
(c)Find P(Hot drink | Pastry).(2)
(d)Explain why P(Pastry | Hot drink) is not equal to P(Hot drink | Pastry), even though both use the same 54 customers who bought both.(2)
(Total for Question 4 is 7 marks)
5
For events A and B, P(B) = 0.6 and P(A|B) = 0.45.
(a)State the formula linking P(A and B), P(B) and P(A|B).(1)
(b)Use the formula to find P(A and B).(2)
(c)Given also that P(A) = 0.3, determine, giving a reason, whether events A and B are independent.(2)
(Total for Question 5 is 5 marks)
6
120 sixth-form students were asked whether they study Mathematics and whether they study Physics. The two-way table shows some of the results.
PhysicsNo PhysicsTotal
Maths3672
No Maths24
Total60120
(a)Complete the two-way table.(3)
(b)Find P(Physics) and P(Physics | Maths).(3)
(c)Hence determine, with a reason, whether studying Mathematics and studying Physics are independent for these students.(2)
(Total for Question 6 is 8 marks)
7
In a survey of 50 students, A is the event that a student plays a musical instrument and B is the event that a student is in a sports team. The Venn diagram shows the number of students in each region.
Total = 50 students A B 12 8 15 15 A = plays a musical instrument     B = is in a sports team
(a)Write down n(A and B).(1)
(b)Find P(A).(1)
(c)Find P(A | B).(2)
(d)Find P(B | A).(2)
(e)State, giving a reason, whether events A and B are independent.(2)
(Total for Question 7 is 8 marks)
8
A student wants to investigate whether Year 11 students who complete at least 4 hours of revision per week are more likely to achieve grade 7 or above in a mock Statistics exam.
(a)Write a suitable hypothesis for this investigation.(1)
(b)The student initially plans to collect data only from her own Statistics class of 28 students. Give one reason why this sampling method could make her conclusions unreliable, and suggest one improvement.(2)
(c)The student instead surveys 80 Year 11 students from across the school, recording whether each completed at least 4 hours of revision per week and whether they achieved grade 7 or above. The two-way table shows the results.
Grade 7+Below grade 7Total
At least 4 hours211435
Fewer than 4 hours93645
Total305080
Find P(Grade 7+ | At least 4 hours) and P(Grade 7+ | Fewer than 4 hours).(4)
(d)Using your answers to part (c), comment on whether the data supports the hypothesis in part (a), and give one limitation of this conclusion.(2)
(Total for Question 8 is 9 marks)
9
A bag contains 10 counters. n of the counters are red and the rest are blue. Priti takes a counter at random from the bag and does not replace it. She then takes a second counter at random from the bag. Given that P(both counters are red) = 1/3.
(a)Show that n(n - 1) = 30.(2)
(b)Hence show that n2 - n - 30 = 0, and solve this equation to find the value of n.(3)
(Total for Question 9 is 5 marks)
10
For statements (i) to (iii), write down whether each is True or False. For part (iv), select the correct option.
(i)If P(A|B) = P(A), then events A and B are independent.(1)
(ii)P(A|B) is always equal to P(B|A).(1)
(iii)For any two events A and B, P(A and B) = P(A) x P(B).(1)
(iv)Which expression is equal to P(A|B)?(1)
  • A) P(A and B) / P(A)
  • B) P(A and B) / P(B)
  • C) P(A) x P(B)
  • D) P(A) + P(B) - P(A and B)
(Total for Question 10 is 4 marks)
11
A pack of 15 cards is numbered 1 to 15. Two cards are drawn at random, one after another, without replacement.
(a)Find the probability that both cards drawn show multiples of 5.(3)
(b)Given that the first card drawn shows a multiple of 5, find the probability that the second card drawn also shows a multiple of 5.(2)
(c)Find the probability that neither card drawn shows a multiple of 5.(2)
(Total for Question 11 is 7 marks)
12
In a survey of 42 students, C is the event that a student has a pet cat and D is the event that a student has a pet dog. The Venn diagram shows the number of students in each region, where x is a positive whole number.
Total = 42 students C D 14 x 2x 10 C = has a pet cat     D = has a pet dog
(a)Use the fact that the four regions in the Venn diagram total 42 to form and solve an equation in x.(2)
(b)Find P(C | D).(2)
(c)Find P(D | C'), where C' means 'does not have a pet cat'.(3)
(Total for Question 12 is 7 marks)
13
A drawer contains 12 batteries, of which 5 are faulty. Two batteries are selected at random, one after another, without replacement.
(a)Find the probability that neither battery selected is faulty.(2)
(b)Hence find the probability that at least one of the two batteries is faulty.(2)
(c)Find the probability that exactly one of the two batteries is faulty.(3)
(Total for Question 13 is 7 marks)
14
200 patients at a clinic were tested for a virus. The total number of patients who tested positive is 3x. Of these, 24 had previously been vaccinated. In total, 80 of the 200 patients had been vaccinated. Given that P(Vaccinated | Positive) = 2/5.
(a)Use the given conditional probability to form an equation in x, and hence find the number of patients who tested positive.(3)
(b)Complete the two-way table for all 200 patients, using your answer to part (a).
PositiveNegativeTotal
Vaccinated2480
Not vaccinated
Total60200

Fill in every blank cell.(3)
(c)Find P(Positive | Not vaccinated).(2)
(Total for Question 14 is 8 marks)
15
A factory buys a component from two suppliers, Supplier X and Supplier Y. 65% of the components come from Supplier X and the rest come from Supplier Y. 3% of the components from Supplier X are faulty, and 7% of the components from Supplier Y are faulty. A component is selected at random. The diagram shows the blank structure of a tree diagram for this situation.
Supplier Outcome
(a)Draw and label a tree diagram for this information, and use it to find the probability that the component is faulty.(3)
(b)Given that the component is faulty, find the probability that it came from Supplier Y.(3)
(Total for Question 15 is 6 marks)
16
A biased spinner can only land on red or blue. The probability that the spinner lands on red is p. The spinner is spun twice; the two spins are independent.
(a)Explain why P(both spins land on red) = p2.(1)
(b)Show that P(both spins land on the same colour) = 2p2 - 2p + 1.(3)
(c)Given that P(both spins land on the same colour) = 13/25, and that the spinner is biased so that red is more likely than blue, form and solve an equation to find the value of p.(4)
(Total for Question 16 is 8 marks)
Mark scheme · H15 Conditional Probability

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16