NCEA Level 2Probability treesAS 91267

Probability trees: 7 exam questions solved step by step

What this lesson covers

  • Each branch of a tree carries a probability conditional on the branches before it.
  • Working backwards through a tree finds the probability of an earlier event given a later one.
Key idea

7 questions, one tree method

  • 1. Draw the tree from the words.
  • 2. Along a path: multiply.
  • 3. Across paths: add.
  • 4. Backwards: path ÷ total.
Your turn

Question 1 · Achieved, in context

  • Ferry: wind warning on 30% of days.
  • Warning: late 60%. No warning: late 10%.
  • Draw the tree. Find P(warning and late).
Worked example

Q1: build the tree

  • First: wind 0.3, calm 0.7.
  • After wind: late 0.6, on time 0.4.
  • After calm: late 0.1, on time 0.9.
Q1: build the tree
Worked example

Q1: P(warning and late)

  • Follow one path.
  • Multiply along it.
  1. W: warning, L: late
  2. P(W and L) = P(W) × P(L | W)
  3. = 0.3 × 0.6multiply along
  4. = 0.18
Your turn

Question 2 · Achieved

  • Same ferry tree.
  • Find P(the ferry is late).
Question 2 · Achieved
Worked example

Q2: P(late)

  • Two paths end in late.
  • Add across them.
  1. P(W and L) = 0.3 × 0.6 = 0.18
  2. P(not W and L) = 0.7 × 0.1 = 0.07
  3. P(L) = 0.18 + 0.07add across
  4. = 0.25
  5. 0.12 + 0.63 = 0.75 = 1 − 0.25
Your turn

Question 3 · Merit

  • Same ferry tree.
  • The ferry is late today.
  • Find P(there was a warning).
Question 3 · Merit
Worked example

Q3: P(W | L)

  • Given late: late paths only.
  • Path ÷ reduced total.
  1. P(W | L) = P(W and L) ÷ P(L)
  2. = 0.18 ÷ 0.25reduced total: P(L)
  3. = 0.72
  4. P(L | W) = 0.6forwards, not backwards
  5. 72% of late days had a warning
Your turn

Question 4 · Merit, in context

  • Kiwifruit from orchards A 50%,
  • B 30% and C 20%.
  • Rejected: A 4%, B 6%, C 10%.
  • Find P(a kiwifruit is rejected).
Worked example

Q4: three first branches

  • First: A 0.5, B 0.3, C 0.2.
  • Then reject or OK: each pair adds to 1.
Q4: three first branches
Worked example

Q4: P(rejected)

  • Three paths end in reject.
  • Multiply along, add across.
  1. A, R: 0.5 × 0.04 = 0.02
  2. B, R: 0.3 × 0.06 = 0.018
  3. C, R: 0.2 × 0.1 = 0.02
  4. P(R) = 0.02 + 0.018 + 0.02add across
  5. = 0.058
  6. about 5.8% of fruit is rejected
Your turn

Question 5 · Merit, in context

  • Same kiwifruit tree.
  • A kiwifruit is rejected.
  • Find P(it came from orchard C).
  • Compare with C's 20% share.
Question 5 · Merit, in context
Worked example

Q5: P(C | R)

  • Reduced total: P(R).
  • Answer in words.
  1. P(C | R) = P(C and R) ÷ P(R)
  2. = 0.02 ÷ 0.058P(R) from question 4
  3. = 0.344827…calculator
  4. = 0.34484 d.p.
  5. C: 20% of fruit, 34% of rejects
Your turn

Question 6 · Excellence

  • Sausage sizzle: P(buys a sausage) = p.
  • Drink: 40% of sausage buyers, 10% of others.
  • 28% buy a drink. Find p,
  • then P(sausage | drink).
Worked example

Q6: a tree with an unknown

  • First: sausage p, none 1 − p.
  • Then drink or no drink.
Q6: a tree with an unknown
Worked example

Q6: form the equation

  • Two paths end in a drink.
  • Together they are 0.28.
  1. P(S) = p, P(not S) = 1 − pbranches add to 1
  2. P(S and D) = 0.4pmultiply along
  3. P(not S and D) = 0.1(1 − p)multiply along
  4. 0.4p + 0.1(1 − p) = 0.28add across: P(D)
  5. 0.4p + 0.1 − 0.1p = 0.28expand
  6. 0.3p + 0.1 = 0.28collect p terms
Worked example

Q6: solve, then go backwards

  • Check p in the equation.
  • Then path ÷ total.
  1. 0.3p + 0.1 = 0.28
  2. 0.3p = 0.18subtract 0.1
  3. p = 0.18 ÷ 0.3 = 0.6divide by 0.3
  4. check: 0.4 × 0.6 + 0.1 × 0.4
  5. = 0.24 + 0.04 = 0.28
  6. P(S | D) = 0.24 ÷ 0.28path ÷ total
  7. = 0.857142…calculator
  8. = 0.85714 d.p.
Your turn

Question 7 · Excellence, in context

  • Two Auckland buses to school.
  • Each is late with probability p,
  • independently.
  • P(at least one late) = 0.36. Find p.
Worked example

Q7: p on every branch

  • Late p, on time 1 − p.
  • The same at both stages.
Q7: p on every branch
Worked example

Q7: use the complement

  • At least one = 1 − none.
  • A probability is 0 to 1.
  1. P(neither late) = (1 − p)²on time × on time
  2. (1 − p)² = 1 − 0.36 = 0.64complement
  3. 1 − p = 0.8 or 1 − p = −0.8
  4. p = 0.2
  5. p = 1.8a probability is ≤ 1
  6. check: 1 − 0.8² = 1 − 0.64 = 0.36
  7. each bus is late 20% of the time
Summary

The method that solved all seven

  • Along, multiply. Across, add. Backwards, divide.
StepHow
Drawfirst event, then conditional
Each setof branches adds to 1
Alongmultiply
Acrossadd the paths
Backwardspath ÷ the reduced total
Unknownform an equation, then check

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