Mixed Probability Problem Set
A mixed probability problem set brings together every KS3 probability skill in one collection of questions, rather than testing a single technique in isolation: single-event probabilities using the 0 to 1 scale, systematically listing outcomes, estimating probability from experimental (relative frequency) data, reading two-way tables and frequency trees, using Venn diagrams and set notation, and multiplying along tree diagrams for independent events. Because a real exam paper never labels each question with the technique it needs, the key extra skill this kind of question tests is recognising, from the wording and the information given, which method applies before starting to calculate, and then working accurately and checking the answer makes sense as a probability.
Before you start
Make sure you're comfortable with these topics first:
Method
- Before calculating anything, read the question fully and identify which probability technique it needs: a single event, listing every outcome, reading data from a table, using given experimental results, a Venn diagram with set notation, or a tree diagram for combined events.
- Look for the specific words that signal each technique: 'estimate the probability' or 'relative frequency' points to experimental data; 'and', 'independent' or 'both' between two separate events usually points to a tree diagram and multiplying; a table of two categories points to a two-way table; words like 'only' or 'neither' point to a Venn diagram.
- For any calculated probability, always double check it is a value between 0 and 1 inclusive (or a percentage between 0% and 100%); a probability outside this range shows a mistake has been made somewhere in the working.
- When a question gives a table, a diagram or written data, extract the exact numbers needed for that part before starting the calculation, since mixed questions often include extra numbers that are not needed for every part.
- For a multi-part question, use an earlier answer (such as a completed two-way table or a value found in part (a)) to answer a later part, rather than recalculating everything from the original data each time.
- Where a question could be solved by more than one valid method, choose the representation already given in the question rather than redrawing it in a different form, to save time.
- Finish by checking probabilities that should be complementary (an event and its opposite) add to 1, and probabilities of every branch leaving one point on a tree diagram add to 1.
Worked example
50 students in Year 9 were asked whether they have a part-time job and whether they own a pet. 18 students have a job only, 21 own a pet only, 6 have both a job and a pet, and the rest have neither. (a) Work out how many students have neither a part-time job nor a pet. (b) A student is picked at random from all 50. Find the probability that the student owns a pet (whether or not they also have a job), giving your answer as a fraction in its simplest form.
- Recognise this is a Venn diagram / set notation question, since it describes 'only' and 'both' categories for two overlapping groups.
- Add the three known regions to find how many students have at least one of the two things: 18 + 21 + 6 = 45.
- Subtract this from the total to find how many have neither: 50 - 45 = 5, which answers part (a).
- For part (b), identify that 'owns a pet' includes both the pet-only region and the both region: 21 + 6 = 27 students.
- Write this as a probability out of the total number of students: 27/50.
- Check whether 27/50 simplifies: 27 = 3 x 3 x 3 and 50 = 2 x 5 x 5, which share no common factor, so 27/50 is already in its simplest form.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1A bag contains 5 red counters, 3 blue counters and 2 green counters, all equally likely to be picked. A counter is picked at random. Find the probability that it is NOT green.Show answer
Answer: 4/5 (8/10, since 8 of the 10 counters are not green)
Q2A cafe menu has 3 types of sandwich (cheese, ham, tuna) and 2 types of drink (juice, water). Using a systematic list, write down all the different sandwich-and-drink combinations a customer could order, and state how many different combinations there are.Show answer
Answer: Cheese+juice, cheese+water, ham+juice, ham+water, tuna+juice, tuna+water - 6 different combinations in total (3 x 2).
Q3A drawing pin is dropped 150 times. It lands point-up 54 times and point-down the rest of the time. Estimate the probability that the drawing pin lands point-down, giving your answer as a decimal.Show answer
Answer: 0.64 (96/150, since the pin landed point-down 150 - 54 = 96 times)
Q4A two-way table shows 90 students' answers to whether they have a packed lunch or a school dinner, and whether they also bring a snack. 24 packed-lunch students bring a snack, 11 packed-lunch students do not, 19 school-dinner students bring a snack, and the rest of the 90 students have a school dinner and do NOT bring a snack. Find the probability that a student picked at random has a school dinner and does NOT bring a snack, giving your answer as a fraction in its simplest form.Show answer
Answer: 2/5 (36/90, since 90 - 35 packed-lunch students - 19 school-dinner-with-snack students = 36 school-dinner-without-snack students)
Q5The probability that a particular train service runs late on any given day is 0.15, and this is independent from day to day. Amir catches this train service on two specific days. Find the probability that the train is late on BOTH days.Show answer
Answer: 0.0225 (0.15 x 0.15)
Q6Out of 40 students, a Venn diagram shows that 13 own a games console only, 9 own a tablet computer only, and 5 own both a games console and a tablet computer. Find the probability that a student picked at random owns NEITHER a games console nor a tablet computer, giving your answer as a fraction in its simplest form.Show answer
Answer: 13/40 (40 - 27 = 13 students own neither, out of 40)
Q7A probability question states: 'Event A and Event B are independent. Find the probability that both A and B happen.' Explain which technique this wording points to, and state the general rule you would use.Show answer
Answer: The word 'independent' together with 'both... happen' points to a tree-diagram-style calculation: for independent events, the probability that both happen is found by multiplying their individual probabilities together, P(A and B) = P(A) x P(B).
Q8Over 200 games, a football team wins 120 times. Based on this, estimate the probability that the team wins their next game. If the team plays 2 more games, and results are assumed independent with this same probability, find the probability that they win BOTH of the next two games.Show answer
Answer: 0.36 - the estimated probability of winning one game is 120/200 = 0.6, so P(win both) = 0.6 x 0.6 = 0.36
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
180 members of a local library were asked whether they borrow fiction books and whether they borrow non-fiction books. 70 borrow both fiction and non-fiction. 55 borrow fiction only. 40 borrow non-fiction only. The rest borrow neither (they use the library for other reasons, such as reading newspapers or using the computers). (a) Work out how many of the 180 members borrow neither fiction nor non-fiction books. (b) A member is picked at random from all 180. Find the probability that they borrow fiction (whether or not they also borrow non-fiction), giving your answer as a fraction in its simplest form. (c) Of the members who borrow non-fiction (whether or not they also borrow fiction), what fraction also borrow fiction? Give your answer in its simplest form.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 6 available
A biased four-sided spinner is tested by spinning it 80 times. It lands on Yellow 24 times. (a) Use this data to estimate the probability that the spinner lands on Yellow on a single spin. (b) The spinner is then spun twice more, and each spin is independent of the others. Using your estimated probability from part (a), find the probability that the spinner lands on Yellow on BOTH of these two spins.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 4 available
Free printable worksheet
Want more practice on paper? Download the mixed probability problem set worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 28 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
Next topics
Not quite what you needed?
Tell us what is missing on mixed probability problem set, or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.