Independent or exclusive: 7 exam questions solved step by step

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What this lesson covers

  • Mutually exclusive events cannot happen together; independent events do not affect each other.
  • Each has its own test, and each test must be written as a probability statement.
Key idea

7 questions, two tests

  • 1. Exclusive: P(A and B) = 0.
  • 2. Independent: P(A and B) = P(A) × P(B).
  • 3. Or: P(A | B) = P(A).
  • 4. Explain it in context.
Your turn

Question 1 · Achieved

  • At a bakery: P(buys a pie) = 0.3,
  • P(pays by card) = 0.8, P(both) = 0.24.
  • Independent? Mutually exclusive?
Worked example

Q1: both tests

  • Multiply, then compare.
  • Then test for 0.
  1. P(pie) = 0.3, P(card) = 0.8
  2. P(pie and card) = 0.24given
  3. P(pie) × P(card) = 0.3 × 0.8
  4. = 0.24
  5. P(pie and card) = P(pie) × P(card)
  6. pie and card are independent
  7. P(pie and card) = 0.24 ≠ 0not mutually exclusive
Your turn

Question 2 · Achieved

  • At a school: P(takes Art) = 0.25,
  • P(Music) = 0.4, P(Art or Music) = 0.5.
  • Are Art and Music mutually exclusive?
Worked example

Q2: find "and" first

  • Addition rule.
  • Test: is "and" 0?
  1. P(A or M) = P(A) + P(M)addition rule
  2. − P(A and M)
  3. 0.5 = 0.25 + 0.4 − P(A and M)
  4. P(A and M) = 0.65 − 0.5 = 0.15
  5. 0.15 ≠ 0some take both
  6. not mutually exclusive
  7. exclusive: P(A or M) = 0.65but it is 0.5
Your turn

Question 3 · Merit

  • 400 shoppers at a supermarket:
  • brought a bag (B), used self-checkout (S).
  • Are B and S independent? Explain.
SelfStaffedTotal
Bag90170260
No bag6080140
Total150250400
Worked example

Q3: the product test

  • Counts into probabilities.
  • Then explain with "given".
  1. P(B and S) = 90 ÷ 400 = 0.225
  2. P(B) = 260 ÷ 400 = 0.65
  3. P(S) = 150 ÷ 400 = 0.375
  4. P(B) × P(S) = 0.65 × 0.375
  5. = 0.24375
  6. 0.225 ≠ 0.24375product test fails
  7. B and S are not independent
  8. P(B | S) = 90 ÷ 150 = 0.6
  9. 0.6 < 0.65self-checkout: fewer bags
Your turn

Question 4 · Merit

  • Seniors at a college: 45% Year 13.
  • 12% of Year 13s and 12% of Year 12s
  • are left-handed. Is being left-handed
  • independent of year level? Explain.
Question 4 · Merit
Worked example

Q4: the conditional test

  • Find P(L) overall.
  • Compare with P(L | Y13).
  1. P(L | Y13) = 0.12from the tree
  2. P(L | Y12) = 0.12
  3. P(L) = 0.45 × 0.12 + 0.55 × 0.12
  4. = 0.054 + 0.066 = 0.12
  5. P(L | Y13) = P(L) = 0.12conditional test
  6. Y13 and L are independent
  7. knowing the year changes nothing
Your turn

Question 5 · Merit, in context

  • Year 13s: P(part-time job) = 0.5,
  • P(club sport) = 0.4, P(both) = 0.14.
  • Test both ideas. Explain in context.
Question 5 · Merit, in context
Worked example

Q5: both tests, then context

  • Two tests, two statements.
  • Compare P(S | J), P(S).
  1. P(J and S) = 0.14 ≠ 0not mutually exclusive
  2. P(J) × P(S) = 0.5 × 0.4 = 0.2
  3. 0.2 ≠ 0.14product test fails
  4. J and S are not independent
  5. P(S | J) = 0.14 ÷ 0.5 = 0.28
  6. P(S | no job) = 0.26 ÷ 0.5 = 0.52
  7. 0.28 < P(S) = 0.4job: less club sport
Your turn

Question 6 · Excellence

  • Mere and Josh enter a speech contest.
  • P(Mere places) = 0.4, P(at least one) = 0.7.
  • Find P(Josh places) if the events are
  • (a) independent, (b) mutually exclusive.
Worked example

Q6 (a): independent

  • Let P(J) = x.
  • "And" = product.
  1. P(M) = 0.4, P(J) = x
  2. P(M or J) = 0.4 + x − P(M and J)
  3. P(M and J) = 0.4xindependent
  4. 0.7 = 0.4 + x − 0.4x
  5. 0.3 = 0.6x− 0.4, collect x
  6. x = 0.5
  7. P(M and J) = 0.4 × 0.5 = 0.2
  8. 0.4 + 0.5 − 0.2 = 0.7
Worked example

Q6 (b): mutually exclusive

  • "And" = 0 now.
  • A different answer.
  1. P(M and J) = 0mutually exclusive
  2. 0.7 = 0.4 + x − 0
  3. x = 0.3
  4. independent 0.5, exclusive 0.3
Your turn

Question 7 · Excellence

  • 600 workers' main way to work.
  • Claim: "Bus and drive are independent,
  • because no worker does both."
  • Evaluate the claim.
Main wayWorkers
Bus150
Drive270
Other180
Total600
Worked example

Q7: test the claim

  • No overlap: exclusive.
  • Now the product test.
  1. P(B) = 150 ÷ 600 = 0.25
  2. P(D) = 270 ÷ 600 = 0.45
  3. P(B and D) = 0 ÷ 600 = 0mutually exclusive
  4. P(B) × P(D) = 0.25 × 0.45
  5. = 0.1125
  6. 0 ≠ 0.1125product test fails
  7. P(D | B) = 0, but P(D) = 0.45
  8. claim not supported
Worked example

Q7: why this always happens

  • Any A and B with P > 0.
  • Exclusive is dependent.
  1. P(A) > 0 and P(B) > 0
  2. so P(A) × P(B) > 0
  3. exclusive: P(A and B) = 0
  4. P(A and B) ≠ P(A) × P(B)
  5. exclusive means not independent
Summary

The two tests that solved all seven

  • Two ideas, two tests, each a probability statement.
StepHow
ExclusiveP(A and B) = 0
IndependentP(A and B) = P(A) × P(B)
Or, givenP(A | B) = P(A)
Write itas a probability statement
Explaincompare P(A | B) with P(A)
Never bothexclusive means dependent

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