NCEA Level 2Probability rulesAS 91267

Probability: when to add and when to multiply

What this lesson covers

  • Probabilities are multiplied for "and" along a path, and added for "or" across outcomes that cannot happen together.
  • A probability is always between 0 and 1, which is a check on every answer.
Key idea

Add or multiply?

  • The examiner's first reason for Not Achieved.
  • Mixing up adding and multiplying.
  • One picture sorts it out.
Key idea

The words decide

  • "And" usually means multiply.
  • "Or" usually means add.
  • "Not" and "at least one" use 1 minus.
EventDoWhen
A and Bmultiplyif independent
A or Baddif they cannot both happen
Not A1 − P(A)the complement
At least one1 − P(none)
Worked example

Probability as area

  • Bus late: 0.2 of the width.
  • Train late: 0.1 of the height.
  • Both late: the corner, 0.2 × 0.1 = 0.02.
Probability as area
Worked example

"Exactly one": add two separate cells

  • Bus late, train on time: 0.2 × 0.9 = 0.18.
  • Bus on time, train late: 0.8 × 0.1 = 0.08.
  • Separate cells, so add: 0.26.
"Exactly one": add two separate cells
Worked example

All the working

  • Not: 1 minus.
  • And: multiply. Or: add.
  1. bus on time: 1 − 0.2 = 0.8not late
  2. train on time: 1 − 0.1 = 0.9not late
  3. both: 0.2 × 0.1 = 0.02and: multiply
  4. neither: 0.8 × 0.9 = 0.72and: multiply
  5. at least one: 1 − 0.72 = 0.28
  6. exactly one: 0.2 × 0.9 + 0.8 × 0.1
  7. = 0.18 + 0.08 = 0.26or: add
  8. check: 0.02 + 0.26 + 0.72 = 1
Key idea

Independent and mutually exclusive

  • Independent: one does not affect the other.
  • Mutually exclusive: they cannot both happen.
  • Different ideas, different operations.
WordMeansSo
Independentno effect on each othermultiply
Mutually exclusivecannot happen togetheradd
Key idea

The "at least one" shortcut

  • Three cells, or one.
  • Use the complement.
  1. long way: 0.02 + 0.18 + 0.08three cells
  2. = 0.28
  3. short way: 1 − 0.72one cell: neither
  4. = 0.28the same
Key idea

Every answer between 0 and 1

  • Adding when you should multiply can pass 1.
  • A probability above 1 is always wrong.
  • The examiner lists this for Not Achieved.
Key idea

Keep four decimal places

  • Rounding early spreads error through a question.
  • Keep at least four decimal places in working.
  • Round only the final answer.
Key idea

Where the marks are

  • Achieved: know when to multiply and when to add.
  • Merit: several steps, like exactly one.
  • Every answer in context.
Summary

What to take away

  • And multiplies, or adds, not subtracts from 1.
Andmultiply (independent events)
Oradd (separate outcomes)
Not1 minus
At least one1 − P(none)
Check0 ≤ p ≤ 1, and totals add to 1
Accuracy4 decimal places in working
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Draw the square if it helps.
  • Check your answers add to 1.
Question
1P(rain) = 0.3 each day, independent. P(rain on both days)?
2P(no rain on either day)?
3P(rain on at least one day)?
4P(rain on exactly one day)?
Show answer

1. 0.3 × 0.3 = 0.09. 2. 0.7 × 0.7 = 0.49. 3. 1 − 0.49 = 0.51. 4. 0.3 × 0.7 + 0.7 × 0.3 = 0.42. Check: 0.09 + 0.42 + 0.49 = 1.

Worked example

The answers: questions 1 and 2

  1. 1. P(rain and rain) = 0.3 × 0.3and: multiply
  2. = 0.09
  3. 2. P(dry) = 1 − 0.3 = 0.7complement
  4. P(dry and dry) = 0.7 × 0.7and: multiply
  5. = 0.49
Worked example

The answers: questions 3 and 4

  1. 3. P(at least one) = 1 − P(none)
  2. = 1 − 0.49
  3. = 0.51
  4. 4. rain, dry or dry, raintwo separate ways
  5. 0.3 × 0.7 + 0.7 × 0.3
  6. = 0.21 + 0.21 = 0.42or: add
  7. check: 0.09 + 0.42 + 0.49 = 1

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