Probability Spaces: Sample Spaces, Events and Axioms · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Sample spaces, events and the three axioms

Mathematics · Probability · ages 18-19
Name ______________________   Date ____________

  1. P(A) = 0.6, P(B) = 0.35 and P(A and B) = 0.2. Find P(A or B).

    Answer: ______________

  2. A menu has 3 starters and 5 mains, and one of each is chosen. How many outcomes are in the sample space?

    • 8
    • 12
    • 15
    • 30

  3. A coin is tossed and a six-sided die is rolled. How many outcomes are in the sample space?

    • 6
    • 12
    • 36

  4. The chance of rain tomorrow is 0.35. What is the chance it does not rain?

    Answer: ______________

  5. Two events A and B overlap. Which statement gives P(A or B)?

    • P(A) + P(B), because the overlap is counted once by adding
    • P(A) + P(B) - P(A and B), because the overlap was counted twice
    • P(A) + P(B) + P(A and B), because the overlap needs extra weight
    • P(A) × P(B), because both events are involved

  6. A die is rolled. Event E is an even roll, {2, 4, 6}. Which set is the complement of E?

    • {2, 4, 6}
    • {1, 2, 3}
    • {1, 3, 5}

  7. Sam assigns P(A) = 0.6, P(B) = 0.7 and P(A or B) = 1.5. Sam says the axioms allow any number between 0 and 2. Is Sam right?

    Circle one:   True   False

  8. Exactly one of outcomes a, b and c must happen. Someone gives them chances 0.1, 0.5 and 0.2 and says no axiom is broken, since each chance is between 0 and 1. Is that right?

    Circle one:   True   False

  9. P(A) = 0.5, P(B) = 0.4 and P(both) = 0.2. Jo skips the overlap and reports 0.9. What is the correct P(A or B)?

    Answer: ______________

  10. Dee says the sample space for rolling two dice has 12 outcomes. Is Dee right?

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Sample spaces, events and the three axioms W1-mt_4gsBeqJtIQ-s1

  1. 0.75 · 0.6 + 0.35 - 0.2 = 0.75, because the overlap was counted twice.
  2. 15 · Each of the 3 starters pairs with each of the 5 mains, so 3 times 5 = 15.
  3. 12 · Two coin faces times six die faces gives 2 times 6, which is 12 possible pairs. Counting possible pairs does not require them to be equally likely.
  4. 0.65 · The complement rule gives 1 minus 0.35, which is 0.65.
  5. P(A) + P(B) - P(A and B), because the overlap was counted twice · Adding P(A) and P(B) counts the overlap twice, so P(A and B) is subtracted once.
  6. {1, 3, 5} · The complement holds every outcome outside E, which is {1, 3, 5}.
  7. False · The axioms force every probability to be at most 1. A union probability of 1.5 would give its complement probability 1 - 1.5 = -0.5, contradicting nonnegativity.
  8. False · No. Each chance is allowed, but the three outcomes cover the whole space, so their chances must total 1. They total 0.8.
  9. 0.7 · Subtract the overlap once: 0.5 plus 0.4 minus 0.2 is 0.7, so Jo double counted.
  10. False · Dee is wrong: 6 faces times 6 faces gives 36 pairs, not 12.
Worksheet · LightMySky