NCEA Level 3Independent or exclusiveAS 91585

Independent or mutually exclusive? The two tests

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

Two ideas, two tests

  • Mixing up the tests: a named 2025 error.
  • Each idea has its own test.
  • Write each test as a probability.
Key idea

What the words mean

  • Mutually exclusive: cannot happen together.
  • Independent: one does not change the other.
  • Overlap versus influence.
WordMeans
Mutually exclusivecannot happen together
Independentno effect on each other
Key idea

The two tests side by side

  • Mutually exclusive: P(A and B) = 0.
  • Independent: P(A and B) = P(A) × P(B).
  • Or: P(A | B) = P(A).
ClaimTest
Mutually exclusiveP(A and B) = 0
IndependentP(A and B) = P(A) × P(B)
orP(A | B) = P(A)
Worked example

Mutually exclusive: no overlap

  • Main way to school, 250 students.
  • P(walk) = 0.32, P(driven) = 0.4.
  • Nobody has two main ways: overlap 0.
Mutually exclusive: no overlap
Worked example

A probability, not a count

  • State it as a probability.
  • Write P(W and D) = 0.
  1. P(neither) = 1 − (0.32 + 0.4)
  2. = 1 − 0.72 = 0.28
  3. no student did botha count, not the test
  4. P(W and D) = 0 ÷ 250 = 0a probability
  5. W and D are mutually exclusive
Worked example

Independence: 400 café customers

  • 160 have a loyalty card.
  • 100 order oat milk; 40 do both.
  • Does a card make a difference?
Oat milkOtherTotal
Loyalty card40120160
No card60180240
Total100300400
Worked example

The rest of the table

  • Each gap: total − known.
  • Check one cell twice.
  1. card, other: 160 − 40 = 120
  2. no card: 400 − 160 = 240
  3. no card, oat: 100 − 40 = 60
  4. other: 400 − 100 = 300
  5. no card, other: 240 − 60 = 180
  6. check: 300 − 120 = 180
Worked example

The product test

  • Find P(L and O).
  • Multiply P(L) × P(O).
  • Equal: independent.
  1. P(L and O) = 40 ÷ 400 = 0.1
  2. P(L) = 160 ÷ 400 = 0.4
  3. P(O) = 100 ÷ 400 = 0.25
  4. P(L) × P(O) = 0.4 × 0.25 = 0.1
  5. P(L and O) = P(L) × P(O)
  6. L and O are independent
Worked example

The conditional test

  • Given a card: 0.25.
  • Overall: also 0.25.
  • No difference.
  1. P(O | L) = 40 ÷ 160 = 0.25card holders only
  2. P(O) = 100 ÷ 400 = 0.25everyone
  3. P(O | L) = P(O)
  4. P(O | no card) = 60 ÷ 240 = 0.25
  5. a card makes no difference
Worked example

A pair that fails the test

  • P(S) = 0.5, P(C) = 0.3.
  • P(S and C) = 0.2.
  • Middle first, then the rest.
A pair that fails the test
Worked example

The product test fails

  • Regions first.
  • Then the product test.
  1. only S = 0.5 − 0.2 = 0.3
  2. only C = 0.3 − 0.2 = 0.1
  3. neither = 1 − 0.6 = 0.41 − (0.3 + 0.2 + 0.1)
  4. P(S) × P(C) = 0.5 × 0.3
  5. = 0.15
  6. P(S and C) = 0.2the overlap
  7. 0.15 ≠ 0.2product test fails
  8. S and C are not independent
Worked example

Explaining it in context

  • Compare with P(C).
  • Then say it in context.
  1. P(C | S) = P(S and C) ÷ P(S)
  2. = 0.2 ÷ 0.5 = 0.4
  3. P(C) = 0.3overall
  4. P(C | S) > P(C)sport: canteen more often
Worked example

Exclusive, so not independent

  • Exclusive: P(W and D) = 0.
  • Product: 0.128, not 0.
  1. P(W and D) = 0mutually exclusive
  2. P(W) × P(D) = 0.32 × 0.4
  3. = 0.128
  4. 0.128 ≠ 0product test fails
  5. W and D are not independent
  6. P(D | W) = 0, but P(D) = 0.4
Key idea

Why they are never the same

  • Both probabilities above zero.
  • Mutually exclusive means not independent.
  • Never use one word for the other.
Key idea

Where the marks are

  • Achieved: the probabilities each test needs.
  • Merit: both tests, as probability statements.
  • Excellence: the relationship, in context.
Summary

What to take away

  • Two ideas, two tests, each written as a probability statement.
ExclusiveP(A and B) = 0
IndependentP(A and B) = P(A)P(B)
orP(A | B) = P(A)
Writeas a probability
Explaincompare P(A | B), P(A)
Not the sameexclusive is dependent
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Classify: independent, exclusive, both, neither.
  • Question 3: explain in context.
GivenFind
1A 0.4, B 0.5, both 0.2classify
2A 0.2, B 0.45, both 0classify
3P(L | R) 0.4, P(L) 0.2independent?
4A 50, B 80, both 20classify
Show answer

1. P(A) × P(B) = 0.2 = P(A and B): independent; P(A and B) ≠ 0, so not mutually exclusive. 2. P(A and B) = 0: mutually exclusive; P(A) × P(B) = 0.09 ≠ 0, so not independent. 3. P(L | R) = 0.4 ≠ P(L) = 0.2: not independent; a student is twice as likely to be late on a rainy day. 4. P(A) = 0.25, P(B) = 0.4, P(A) × P(B) = 0.1 = 20 ÷ 200: independent; not mutually exclusive.

Worked example

The answers: questions 1 and 2

  1. 1. P(A) × P(B) = 0.4 × 0.5 = 0.2
  2. = P(A and B): independent
  3. P(A and B) = 0.2 ≠ 0not mutually exclusive
  4. 2. P(A and B) = 0mutually exclusive
  5. P(A) × P(B) = 0.2 × 0.45 = 0.09
  6. 0.09 ≠ 0not independent
Worked example

The answers: questions 3 and 4

  1. 3. P(L | R) = 0.4, P(L) = 0.2
  2. 0.4 ≠ 0.2: not independent
  3. 0.4 ÷ 0.2 = 2twice the overall chance
  4. 4. P(A) = 50 ÷ 200 = 0.25
  5. P(B) = 80 ÷ 200 = 0.4
  6. P(A) × P(B) = 0.25 × 0.4 = 0.1
  7. P(A and B) = 20 ÷ 200 = 0.1
  8. equal: independent
  9. P(A and B) ≠ 0: not exclusive

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