Loading...

About Probability (Combined Events) — Class 9 IB

Calculate probabilities of combined events using addition and multiplication rules. This topic is part of the IB Class 9 mathematics syllabus (chapter: Unit 9). On this page you can practice 58 questions across three difficulty levels — 20 easy, 18 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Probability (Combined Events) — solved examples for Class 9 IB

Example 1easy

Which of the following scenarios describes two independent events?
  1. A)A) Rolling a standard die and then flipping a coin.
  2. B)B) Drawing two cards from a deck without replacement.
  3. C)C) Picking a blue marble from a bag, not replacing it, and then picking another blue marble.
  4. D)D) The probability of it raining and the probability of needing an umbrella on the same day.

Step-by-step solution

  1. Identify the definition of independent events: events where the occurrence of one does not influence the probability of the other occurring.
  2. Analyze each option: Drawing cards without replacement (B) and picking marbles without replacement (C) are dependent. Rain and needing an umbrella (D) are highly dependent. Rolling a die and flipping a coin are entirely separate actions, so their outcomes do not affect each other.
  3. Therefore, rolling a die and flipping a coin are independent events.

Answer: A) Rolling a standard die and then flipping a coin.

Example 2medium

A fair coin is tossed and a standard six-sided die is rolled. What is the probability of getting a Head on the coin AND an even number on the die?
  1. A)1/4
  2. B)1/2
  3. C)1/6
  4. D)1/12

Step-by-step solution

  1. The probability of getting a Head (H) on a fair coin is P(H) = 1/2.
  2. The even numbers on a standard six-sided die are {2, 4, 6}. So, the probability of getting an even number (E) is P(E) = 3/6 = 1/2.
  3. Since these are independent events, the probability of both occurring is the product of their individual probabilities.
    P(HandE)=P(H)×P(E)=(1/2)×(1/2)=1/4P(H and E) = P(H) × P(E) = (1/2) × (1/2) = 1/4

Answer: 1/4

Example 3hard

A student is preparing for two independent math quizzes, Quiz A and Quiz B. The probability of passing Quiz A is 0.7, and the probability of passing Quiz B is 0.6. What is the probability that the student passes exactly one of the two quizzes?
  1. A)0.46
  2. B)0.88
  3. C)0.42
  4. D)0.58

Step-by-step solution

  1. Calculate the probability of failing each quiz: P(Fail A) = 1 - P(Pass A) = 1 - 0.7 = 0.3. P(Fail B) = 1 - P(Pass B) = 1 - 0.6 = 0.4.
  2. Identify the two mutually exclusive ways to pass exactly one quiz: (Pass A AND Fail B) OR (Fail A AND Pass B).
  3. Since the quizzes are independent, calculate the probability for each scenario: P(Pass A AND Fail B) = P(Pass A) × P(Fail B) = 0.7 × 0.4 = 0.28. P(Fail A AND Pass B) = P(Fail A) × P(Pass B) = 0.3 × 0.6 = 0.18.
  4. Add the probabilities of these two mutually exclusive scenarios to find the total probability of passing exactly one quiz: P(Exactly one pass) = 0.28 + 0.18 = 0.46.

Answer: 0.46

Practice questions on Probability (Combined Events)

  1. Q1.easy

    Consider rolling a standard six-sided die once. Which pair of events is mutually exclusive?
    1. A)A) Rolling an even number and rolling a prime number.
    2. B)B) Rolling a number greater than 4 and rolling an odd number.
    3. C)C) Rolling a multiple of 3 and rolling an even number.
    4. D)D) Rolling a 1 and rolling a 6.
    Show answer

    Answer: D) Rolling a 1 and rolling a 6.

    Hint: Mutually exclusive events cannot happen at the same time. Check if there's any common outcome between the events in each option.

  2. Q2.easy

    A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If one marble is chosen at random, what is the probability of picking a red or a green marble?
    1. A)A) 3/10
    2. B)B) 7/10
    3. C)C) 1/2
    4. D)D) 4/5
    Show answer

    Answer: B) 7/10

    Hint: Since picking a red marble and picking a green marble are mutually exclusive events, you can use the simple addition rule.

  3. Q3.easy

    In a class of 30 students, 15 play football, 10 play basketball, and 5 students play both football and basketball. If a student is chosen randomly, what is the probability that the student plays football or basketball?
    1. A)A) 1/2
    2. B)B) 1/3
    3. C)C) 5/6
    4. D)D) 2/3
    Show answer

    Answer: D) 2/3

    Hint: These events are not mutually exclusive because some students play both sports. Remember to subtract the overlap to avoid double-counting.

  4. Q4.medium

    From a well-shuffled deck of 52 playing cards, a single card is drawn. What is the probability that the card drawn is a King OR an Ace?
    1. A)1/13
    2. B)2/13
    3. C)3/26
    4. D)1/26
    Show answer

    Answer: 2/13

    Hint: Consider if drawing a King and drawing an Ace can happen at the same time. If not, they are mutually exclusive events.

  5. Q5.medium

    Two friends, A and B, are attempting to solve a math problem. The probability that A solves the problem is 0.7, and the probability that B solves the problem is 0.6. Assuming their attempts are independent, what is the probability that at least one of them solves the problem?
    1. A)0.88
    2. B)0.42
    3. C)0.18
    4. D)0.58
    Show answer

    Answer: 0.88

    Hint: It's often easier to calculate the probability that *neither* solves the problem and then use the complementary event rule.

  6. Q6.medium

    Which of the following scenarios describes two dependent events?
    1. A)Flipping a coin and rolling a die.
    2. B)Drawing a card from a deck, not replacing it, and then drawing a second card.
    3. C)Tossing two coins simultaneously.
    4. D)Rolling a die twice.
    Show answer

    Answer: Drawing a card from a deck, not replacing it, and then drawing a second card.

    Hint: Dependent events are those where the outcome of the first event affects the probability of the second event.

  7. Q7.hard

    In a class of 40 students, 25 students play Football, and 20 students play Basketball. If 8 students play neither sport, what is the probability that a randomly selected student plays exactly one of the two sports?
    1. A)13/40
    2. B)19/40
    3. C)27/40
    4. D)32/40
    Show answer

    Answer: 19/40

    Hint: First, find out how many students play at least one sport. Then use the formula for non-mutually exclusive events to find the number of students who play both sports.

  8. Q8.hard

    A bag contains 5 red, 4 blue, and 3 green marbles. Three marbles are drawn one after another without replacement. What is the probability that the first marble is red, the second is blue, and the third is red?
    1. A)5/132
    2. B)5/66
    3. C)2/33
    4. D)4/33
    Show answer

    Answer: 2/33

    Hint: This is a problem involving dependent events. Remember that the total number of marbles and the number of specific colored marbles change after each draw.

  9. Q9.hard

    A factory produces light bulbs. The probability that a single bulb is defective is 0.05. If a quality inspector randomly selects 4 bulbs from a large batch (assume independence), what is the probability that at least one of them is defective?
    1. A)0.050
    2. B)0.185
    3. C)0.815
    4. D)0.190
    Show answer

    Answer: 0.185

    Hint: When 'at least one' is involved, it's often simpler to calculate the probability of the complementary event, which is 'none' of them being defective.

These are 9 of the 58 questions available for Probability (Combined Events). Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.