The Addition Rule and Mutually Exclusive Events · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Two questions at once, without double counting

Mathematics · Probability · ages 15-16
Name ______________________   Date ____________

  1. True or false: If two events are mutually exclusive, they must also be independent.

    Circle one:   True   False

  2. In probability, what does 'A or B' include?

    • only the cases where exactly one of the two happens
    • only the cases where both of them happen together
    • the cases where A happens, where B happens, or where both do
    • whichever of the two events happens to have the larger probability of the two

  3. An event has probability 0.35. What is the probability that it does not happen?

    Answer: ______________

  4. The probability that a card drawn from a deck is a heart is 0.25. What is the probability that it is NOT a heart?

    Answer: ______________

  5. Leo works out P(heart or picture card) as 13/52 + 12/52 = 25/52. What has he missed?

    • he should have multiplied the two probabilities together instead of adding them
    • he should have used 52 + 52 as the denominator for the two events
    • the three cards that are both a heart and a picture card have been counted twice
    • picture cards and hearts cannot be added because they are different kinds of thing

  6. Two correct probabilities have been added correctly to find P(A or B), giving 1.3. What must have happened?

    • the answer should have been written as 130 percent rather than as a decimal
    • one of the two probabilities must have been measured incorrectly
    • something has been counted twice, since no probability can be above 1
    • the two events must have been mutually exclusive after all

  7. P(A) = 0.4 and P(B) = 0.5. What is the largest P(A or B) could possibly be?

    • 0.5, since that is the larger of the two probabilities involved
    • 0.2, which is what you get by multiplying the two together
    • 0.9, when the two events have no overlap at all between them
    • 1, since any probability is allowed to reach one at the very most

  8. Why is there a subtraction in the addition rule at all?

    • to keep the final answer from ever growing larger than one, whatever the events are
    • because probabilities always have to be reduced before they are reported
    • because subtracting is what turns two separate events into a single one
    • because anything in both events gets counted once in each, so one copy is spare

  9. A stall has 60 mugs. What extra information turns 'chipped' and 'striped' into a question you can answer with the addition rule?

    • how many mugs there are on the stall altogether at the start
    • which of the two groups holds more mugs than the other
    • whether the mugs are sold in any particular order over the course of the day
    • how many mugs are chipped, how many are striped, and how many are both

  10. At a school, P(a student likes pizza) = 0.7, P(a student likes pasta) = 0.4, and P(a student likes pizza or pasta) = 0.85. What is P(a student likes both)?

    Answer: ______________

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Answer key

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

Two questions at once, without double counting W1-mt_QsIusFefeE-s1

  1. False · Not necessarily. If both probabilities are positive, mutually exclusive events are dependent: learning one happened rules out the other. If either probability is 0, the events are independent.
  2. the cases where A happens, where B happens, or where both do · In mathematics 'or' takes in the both case too. That is exactly why the overlap has to be subtracted rather than left out.
  3. 0.65 · Happening and not happening cover everything and cannot both occur, so they add to 1. That makes the answer 1 - 0.35 = 0.65.
  4. 0.75 · All the outcomes in a sample space add up to 1, so the two complementary events must add up to 1 too.
  5. the three cards that are both a heart and a picture card have been counted twice · The jack, queen and king of hearts appear in both of Leo's counts. Taking one copy of them off gives 22/52 rather than 25/52.
  6. something has been counted twice, since no probability can be above 1 · The inputs and addition are correct. Since P(A or B) cannot exceed 1, a sum of 1.3 must include an overlap counted twice. Subtract that overlap once; its probability is at least 0.3.
  7. 0.9, when the two events have no overlap at all between them · The overlap can be anywhere from 0 up to 0.4, and P(A or B) is largest when the overlap is smallest. With no overlap at all it is 0.4 + 0.5 = 0.9.
  8. because anything in both events gets counted once in each, so one copy is spare · The subtraction removes the second copy of the outcomes that belong to both events, so each one is counted once.
  9. how many mugs are chipped, how many are striped, and how many are both · The rule needs all three counts. Without the overlap, the two separate counts cannot be combined without either double counting or guessing.
  10. 0.25 · The overlap is the separate probabilities added together minus the union: 0.7 + 0.4 - 0.85 = 0.25.
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