University of the Philippines College Admission Test · Mathematics

Statistics, Probability and Combinatorics

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

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

All 35 questions

  1. How many distinct ways can a committee of 3 members be selected from a group of 8 candidates?
  2. In how many ways can 4 distinct executive positions be assigned to 4 individuals chosen from 7 applicants?
  3. In how many distinct ways can 6 people be seated around a circular conference table?
  4. How many distinct permutations can be formed from all the letters of the word 'PHILIPPINES'?
  5. A card is drawn from a standard 52-card deck. What is the probability of drawing a face card (Jack, Queen, or King)?
  6. A fair coin is tossed 3 times. What is the probability of obtaining at least two heads?
  7. A bag contains 5 red balls and 7 blue balls. Two balls are drawn successively without replacement. What is the probability that both are red?
  8. Two fair six-sided dice are rolled. What is the probability that the sum of the numbers shown is equal to 8?
  9. For the dataset {4, 7, 8, 8, 9, 12, 15}, what is the difference between the mean and the median?
  10. What is the interquartile range (IQR) of the dataset {3, 5, 7, 8, 9, 11, 15, 16, 20}?
  11. What is the population variance of the dataset {2, 4, 6, 8, 10}?
  12. In how many ways can a committee of 3 boys and 2 girls be chosen from 6 boys and 5 girls?
  13. The probability that a candidate passes the entrance exam is 3/4. If 3 candidates take the test independently, what is the probability that at least one passes?
  14. A disease has a prevalence of 2% in a population. A diagnostic test has 90% sensitivity and 95% specificity. If a person tests positive, what is the probability they actually have the disease?
  15. A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?
  16. In a carnival game, a player rolls a fair die. The player wins ₱60 if a 6 is rolled, wins ₱12 if a 5 is rolled, and loses ₱12 if any other number is rolled. What is the player's expected payoff per ro…
  17. A student scores 82 on an exam where the class mean is 70 and the standard deviation is 8. What is the student's z-score?
  18. Exam scores are normally distributed with mean 500 and standard deviation 50. According to the Empirical Rule, what percentage of examinees score between 400 and 600?
  19. What is the minimum number of students required in a room to guarantee that at least two share the same birth month?
  20. If P(A) = 0.6, P(B) = 0.5, and P(A intersect B) = 0.3, what is the probability that neither event A nor event B occurs?
  21. At an academic assembly, 12 scholars meet and every scholar shakes hands with every other scholar exactly once. How many total handshakes occur?
  22. In how many ways can 4 math books and 3 science books be arranged on a shelf if all math books must stay together and all science books must stay together?
  23. A dart is thrown at a square board of side 10 cm. If it lands randomly on the board, what is the probability that it lands inside an inscribed circle of diameter 10 cm?
  24. A commuter travels to campus at 40 km/h and returns along the same route at 60 km/h. What is the average speed for the round trip?
  25. A dataset has a mean of 40 and a standard deviation of 6. What is its coefficient of variation?
  26. A box contains 4 blue and 6 green marbles. A marble is drawn, its color recorded, and returned to the box. A second marble is then drawn. What is the probability that both are green?
  27. How many 4-digit odd numbers can be formed using the digits {1, 2, 3, 4, 5} without repetition?
  28. A discrete random variable X has probability distribution P(X = 1) = 0.15, P(X = 2) = 0.35, P(X = 3) = 0.20, and P(X = 4) = k. What is the value of k?
  29. There are 3 highway routes from City A to City B, 4 routes from City B to City C, and 2 routes from City C to City D. How many distinct travel routes exist from City A to City D passing through B and…
  30. In a research survey, selecting every 10th student from an alphabetical roster of registered undergraduates exemplifies which sampling methodology?
  31. Which statistical diagram visually displays the minimum, first quartile, median, third quartile, and maximum of a continuous distribution?
  32. If the constant value 5 is added to every score in a distribution with a standard deviation of 4, what is the standard deviation of the new scores?
  33. If every score in a distribution with standard deviation 4 is multiplied by 3, what is the standard deviation of the transformed scores?
  34. From a squad of 8 candidates including Alex and Bea, a 4-person delegation is formed that must include Alex but exclude Bea. How many distinct delegations can be chosen?
  35. What is the probability of obtaining a total score of at least 11 when rolling two fair six-sided dice?

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