Year 7ProbabilityTheoretical probabilityYear 7 Maths

Listing outcomes, probability and complements

What this lesson covers

  • Lists, tables, and tree diagrams are useful systematic methods for generating all possible outcomes.
  • If all possible outcomes are assumed to be equally likely, the probability of an event is number of ways the event can happen / total number of possible outcomes.
  • Probabilities can be expressed as a fraction or decimal between 0 and 1, or as a percentage between 0% and 100%.
  • An event is a subset of the sample space and thus can be a single outcome or a combination of outcomes.
  • The probability of an event and its complement add to 1.
Key idea

Flip two coins. How many outcomes?

  • Most people say three.
  • Two heads, two tails, or one of each.
  • There are four.
First coinSecond coinOutcome
HeadsHeadsHH
HeadsTailsHT
TailsHeadsTH
TailsTailsTT
Key idea

The problem with thinking of them

  • Thinking of outcomes is how you miss one.
  • You need a method that cannot skip anything.
  • The curriculum calls that being systematic.
First coinSecond coinOutcome
HeadsHeadsHH
HeadsTailsHT
TailsHeadsTH
TailsTailsTT
Key idea

A tree diagram cannot miss one

  • Every branch of the second coin, from every branch of the first.
  • Follow each path from left to right.
  • Four paths. Four outcomes.
A tree diagram cannot miss one
Key idea

Why "one of each" is two outcomes

  • Head then tail is a different result from tail then head.
  • They are separate paths through the tree.
  • Which is why one of each is twice as likely as two heads.
Why "one of each" is two outcomes
Key idea

Turning a list into a probability

  • Count the outcomes you want.
  • Divide by the total number of outcomes.
  • That works when every outcome is equally likely.
Turning a list into a probability
Worked example

P(two heads)

  • Only HH gives two heads. That is one outcome.
  • There are four altogether.
  • So the probability is one quarter.
EventOutcomesChance
Two headsHH1 of 4
One of eachHT, TH2 of 4
At least one headHH, HT, TH3 of 4
No headsTT1 of 4
Key idea

An event can be more than one outcome

  • "Two heads" is a single outcome.
  • "One of each" is two of them, HT and TH.
  • Count all the ones that qualify.
EventOutcomesChance
Two headsHH1 of 4
One of eachHT, TH2 of 4
At least one headHH, HT, TH3 of 4
No headsTT1 of 4
Worked example

P(one of each)

  • Two outcomes qualify: HT and TH.
  • Two out of four.
  • Which simplifies to one half.
P(one of each)
Key idea

Three ways to write the same chance

  • A quarter. Or 0.25. Or 25 percent.
  • All the same number, exactly as in fractions.
  • Use whichever the question uses.
FractionDecimalPercentage
1/40.2525%
1/20.550%
3/40.7575%
1/80.12512.5%
7/80.87587.5%
Key idea

Always between 0 and 1

  • 0 means it cannot happen. 1 means it must.
  • Everything else sits between them.
  • A probability above 1 is a mistake, every time.
Always between 0 and 1
Key idea

Three coins

  • Add a third stage to the tree.
  • Every path from before splits in two again.
  • Eight outcomes.
Three coins
Key idea

Counting without drawing

  • Two choices, then two, then two: 2 × 2 × 2 = 8.
  • A coin and a three-colour spinner: 2 × 3 = 6.
  • Two dice: 6 × 6 = 36.
ExperimentWorkingOutcomes
2 coins2 × 24 outcomes
3 coins2 × 2 × 28 outcomes
Coin and 3-spinner2 × 36 outcomes
2 dice6 × 636 outcomes
Key idea

The opposite of an event

  • Every event has an opposite: it does not happen.
  • That opposite is called its complement.
  • Together they cover every outcome, so they total 1.
EventPComplementP
Two heads1/4not two heads3/4
At least one head7/8no heads1/8
Rolling a six1/6not a six5/6
Key idea

Why they have to add to 1

  • Something has to happen.
  • Either the event does, or it does not.
  • So between them they account for everything.
EventPComplementP
Two heads1/4not two heads3/4
At least one head7/8no heads1/8
Rolling a six1/6not a six5/6
Worked example

Where that saves you work

  • Three coins. What is P(at least one head)?
  • The long way is counting seven outcomes.
  • The short way: the only failure is TTT.
MethodWhat you doWorking
The long waycount 7 outcomesHHH HHT HTH HTT THH THT TTH
The short waythe only failure is TTT1 − 1/8
Both give7/8
Worked example

1 minus one eighth

  • P(no heads) is TTT, which is one out of eight.
  • So P(at least one head) is 1 − 1/8.
  • Seven eighths, or 0.875, or 87.5 percent.
1 minus one eighth
Key idea

Comparing two events

  • Put both on the 0-to-1 scale.
  • Whichever is further right is more likely.
  • Convert to the same form first if you need to.
Comparing two events
Your turn

Your turn: four questions

  • Pause here and try all four.
  • Do question 1 as a tree, not from memory.
  • Then use it for the rest.
Question
1Two coins are flipped. List every possible outcome.
2What is the probability of at least one head?
3Write that answer as a decimal and a percentage.
4What is the probability of no heads? Check it against question 2.
Show answer

1. HH, HT, TH, TT: four of them. 2. 3/4. Three of the four outcomes contain at least one head. 3. 0.75 and 75%. 4. 1/4, only TT. And 3/4 + 1/4 = 1, which is the complement check.

Worked example

The answers

  • HH, HT, TH and TT: four outcomes.
  • Three contain a head, so 3/4, which is 0.75 or 75%.
  • No heads is 1/4, and 3/4 + 1/4 = 1.
Answer
1HH, HT, TH, TT
23/4
30.75 and 75%
41/4, and the pair totals 1
Summary

What to take away

  • List outcomes with a tree, not from memory.
  • Probability is ways it happens over total outcomes.
  • And P(not A) is 1 − P(A), which is often far quicker.
Remember
List systematicallya tree cannot miss an outcome
Probabilityways it happens ÷ total outcomes
An eventcan be one outcome or several
Fraction, decimal, percentagethe same number three ways
P(not A)is 1 − P(A), often much quicker

Is your child on track?

Take the free placement test