UCAT · Decision Making

Probability, sets and arguments

38 practice questions, each with the answer and an explanation.

Practise Probability, sets and arguments now

Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 38 questions

  1. A bag holds 6 red, 6 blue and 3 green counters. Two are taken at random without replacement. What is the probability that both are red?
  2. Of 50 students, 27 have a bicycle, 29 have a scooter and 16 do neither. How many have a bicycle but do not have a scooter?
  3. 4 people (Amira, Grace, Isla, Jamal) are ranked from oldest to youngest. Grace is older than Jamal. Isla is immediately older than Amira. Isla is older than Jamal. Isla is the oldest. Amira is older t…
  4. A bag holds 3 red, 3 blue and 5 green counters. Two are taken at random without replacement. What is the probability that neither is red?
  5. A restaurant argues: 'Sales of our vegetarian dishes rose 20 per cent, so more of our customers must have become vegetarian.' Which fact most undermines the argument?
  6. On an island each person either always tells the truth or always lies. Tom says: "Priya is a truth-teller." Priya says: "Exactly one of the three of us is a liar." Amira says: "Tom and I are the same…
  7. A company's profit rose from 40,000 pounds to 50,000 pounds, then fell by 20 per cent. What is the final profit?
  8. On an island each person either always tells the truth or always lies. Kiran says: "Hugo and Leah are both liars." Hugo says: "Leah and I are the same type." Leah says: "Hugo is a truth-teller." Which…
  9. The mean of five numbers is 12. Four of them are 8, 10, 14 and 16. What is the fifth?
  10. Of 40 students, 28 speak German, 24 speak Italian and 8 do neither. How many speak German but do not speak Italian?
  11. 5 people (Chloe, Jamal, Kiran, Marcus, Rhys) are ranked from fastest to slowest. Jamal is immediately faster than Rhys. Jamal is the fastest. Rhys is immediately faster than Chloe. Chloe is immediatel…
  12. A bag holds 4 red, 7 blue and 2 green counters. Two are taken at random without replacement. What is the probability that one is red and one is blue (in either order)?
  13. Some doctors are researchers. All researchers are curious. Which statement must be true?
  14. Of 100 students, 35 speak German, 41 speak Italian and 45 do neither. How many speak German but do not speak Italian?
  15. Four boxes are labelled P, Q, R and S. P is heavier than Q. R is lighter than Q. S is heavier than P. Which is the lightest box?
  16. If the alarm is armed, then the meeting is held. If the meeting is held, then the report is filed. Given that the alarm is armed, which statement must be true?
  17. 5 people (Dev, Elena, Isla, Leah, Rhys) are ranked from tallest to shortest. Dev is immediately taller than Isla. Isla is taller than Rhys. Leah is taller than Dev. Isla is immediately taller than Ele…
  18. A bag holds 3 red, 4 blue and 2 green counters. Two are taken at random without replacement. What is the probability that one is red and one is blue (in either order)?
  19. Three friends, Asha, Ben and Cal, each own one of a cat, a dog and a rabbit. Asha does not own the dog. Ben does not own the cat or the dog. Who owns the dog?
  20. A bag holds 7 red, 5 blue and 3 green counters. Two are taken at random without replacement. What is the probability that one is red and one is blue (in either order)?
  21. If the screen is lit, then the lights are working. If the lights are working, then the heating runs. It is known that it is not the case that the heating runs. Which statement must be true?
  22. 5 people (Dev, Femi, Kiran, Nia, Rhys) are ranked from fastest to slowest. Dev is immediately faster than Kiran. Rhys is the slowest. Nia is faster than Dev. Dev is faster than Femi. Who is the third…
  23. If the library is open, the café is open. The café is closed. What can be concluded?
  24. A train leaves at 09:40 and the journey takes 2 hours 35 minutes. A passenger needs 20 minutes to travel from the station to a meeting that starts at 13:00. What is the latest train departure that wou…
  25. Of 50 students, 18 speak German, 31 speak Italian and 12 do neither. How many speak German but do not speak Italian?
  26. On an island each person either always tells the truth or always lies. Sana says: "Amira is a liar." Ben says: "Amira is a truth-teller." Amira says: "Ben and I are the same type." Which of the follow…
  27. Which of the following is the best example of a false dichotomy?
  28. If the server is online, then the report is filed. If the report is filed, then the meeting is held. It is known that it is not the case that the meeting is held. Which statement must be true?
  29. 5 people (Hugo, Isla, Jamal, Kiran, Rhys) are ranked from oldest to youngest. Rhys is older than Jamal. Rhys is immediately older than Hugo. Jamal is older than Kiran. Kiran is immediately older than…
  30. On an island each person either always tells the truth or always lies. Hugo says: "At least two of the three of us are liars." Marcus says: "Exactly one of the three of us is a liar." Tom says: "Hugo…
  31. A bag holds 3 red, 3 blue and 4 green counters. Two are taken at random without replacement. What is the probability that neither is red?
  32. On an island each person either always tells the truth or always lies. Marcus says: "Isla is a liar." Isla says: "At least two of the three of us are liars." Kiran says: "Marcus and Isla are both liar…
  33. Of 80 students, 34 play football and 51 play tennis. 12 do both. How many do neither?
  34. 4 people (Amira, Leah, Priya, Zara) are ranked from fastest to slowest. Amira is faster than Priya. Amira is the fastest. Priya is faster than Zara. Leah is immediately faster than Priya. Amira is fas…
  35. A shop sells pens in packs of 6 and packs of 10. A teacher needs exactly 46 pens. Which combination gives exactly 46?
  36. 4 people (Ben, Marcus, Tom, Zara) are ranked from tallest to shortest. Tom is taller than Zara. Marcus is immediately taller than Tom. Ben is taller than Tom. Who is the third tallest?
  37. A study finds that people who own more books earn higher salaries. Which is the most reasonable conclusion?
  38. In a group of 50 people, 30 drink tea, 25 drink coffee and 10 drink neither. How many drink both?

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