Year 8ProbabilityTheoretical probabilityYear 8 Maths

Probability as a fraction, a decimal and a percentage

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

Which is more likely: 3/8 or 40%?

  • You cannot tell while they are written differently.
  • 3/8 is 0.375. And 40% is 0.4. So the 40% is bigger.
  • That was a conversion problem, not a probability one.
WrittenAs a decimal
3/80.375
40%0.4larger
Key idea

Where a probability comes from

  • Count the ways the thing you want can happen.
  • Divide by how many outcomes there are altogether.
  • One face is a 6, out of six faces. So P(6) is 1/6.
StepWorkingGives
Ways it can happenone face is a 61
Total outcomessix faces6
Probability1 ÷ 61/6
Key idea

But only if the outcomes are equally likely

  • That formula assumes no face is favoured over another.
  • For the weighted die from last lesson, it is simply wrong.
  • Check the assumption before you use the formula.
But only if the outcomes are equally likely
Key idea

Every probability sits between 0 and 1

  • 0 means impossible. 1 means certain.
  • Everything real sits somewhere between them.
  • As a percentage that is 0% to 100%. Never more.
Every probability sits between 0 and 1
Key idea

And can be written three ways

  • P(6) is 1/6, which is 0.167, which is 16.7%.
  • One probability. Three notations. Same number.
  • Exactly as fractions, decimals and percentages always were.
EventFractionDecimal or percentage
P(6)1/60.167 or 16.7%
P(even)1/20.5 or 50%
P(more than 4)1/30.333 or 33.3%
P(not 6)5/60.833 or 83.3%
Key idea

Which is not a formality

  • A fraction is exact. 1/3 beats 0.333 every time.
  • A percentage is easiest to say out loud and to compare.
  • Pick whichever the question needs, and convert freely.
NotationBest for
Fractionbeing exact1/3
Decimalcomparing and calculating0.333
Percentagesaying it to somebody33.3%
Key idea

An event can be more than one outcome

  • Rolling a 6 is one outcome. That is an event.
  • Rolling an even number is three outcomes. Also one event.
  • An event is just the set of outcomes that count as a win.
EventOutcomes in itProbability
Rolling a 6one outcome1/6
Rolling an even number2, 4 or 63/6
Rolling more than 45 or 62/6
All of theseare single events
Worked example

So count the ways, not the questions

  • "More than 4" on a die. Which faces count? 5 and 6.
  • Two ways out of six. So 2/6, which is 1/3.
  • Careful: more than 4 does not include the 4.
So count the ways, not the questions
Worked example

A bag of ten counters

  • 5 red, 3 blue, 2 green. Ten altogether.
  • P(red) is 5/10, which is a half, or 50%.
  • P(blue) is 30%, P(green) 20%. They total 100%.
ColourCountProbability
Red5 of 101/2 or 50%
Blue3 of 100.3 or 30%
Green2 of 101/5 or 20%
Together100%
Key idea

And they always total 1

  • Every draw produces exactly one of the three colours.
  • So the three probabilities have to account for everything.
  • 50 plus 30 plus 20 is 100%. It is a useful check.
ColourCountProbability
Red5 of 101/2 or 50%
Blue3 of 100.3 or 30%
Green2 of 101/5 or 20%
Together100%
Worked example

Comparing two events

  • Is red more likely than blue? Put them in one notation.
  • 1/2 is 0.5. 3/10 is 0.3.
  • So red is more likely. That is why converting matters.
EventConvertedValue
P(red) = 1/2as a decimal0.5
P(blue) = 3/10as a decimal0.3
So redis more likely
Key idea

Everything else is the complement

  • P(6) is 1/6. So P(not 6) must be the other 5/6.
  • Together they account for every outcome, so they total 1.
  • P(not A) = 1 − P(A). Always.
EventWorkingProbability
P(6)1/6
P(not 6)5/6
Together1/6 + 5/61
So P(not A)1 − P(A)
Key idea

Two coins, and all four outcomes

  • A tree lists them without missing any.
  • Two branches, then two more from each: four outcomes.
  • HH, HT, TH, TT. Not three, which is the usual guess.
Two coins, and all four outcomes
Worked example

Now: at least one head

  • Count them directly: HH, HT and TH. Three out of four.
  • Or notice that the only failure is TT, which is 1/4.
  • So it is 1 − 1/4, which is 3/4. One subtraction.
Now: at least one head
Key idea

"At least" is the signal

  • Whenever a question says at least, try the complement first.
  • Counting the successes is usually the longer job.
  • And the longer job is where the miscounts happen.
RouteWorkAnswer
P(at least one head)count HH, HT, TH3/4
Or: P(no heads)only TT1/4
So P(at least one)1 − 1/43/4
"At least"means try the complement
Your turn

Your turn: four questions

  • The bag holds 4 red and 6 blue. Pause here.
  • Question 1 wants all three notations.
  • Question 4: read whether the 4 itself is included.
Question
A bag holds 4 red and 6 blue counters.
1Find P(red) as a fraction, a decimal and a percentage.
2Find P(not red).
3Flip two coins. Find P(two tails).
4Roll a die. Find P(more than 4).
Show answer

1. 4/10, which is 2/5, or 0.4, or 40%. 2. 0.6, or 60%, or 3/5. Either count the 6 blue out of 10, or take 1 − 0.4. 3. 1/4. The four outcomes are HH, HT, TH and TT, and only one is two tails. 4. 2/6, which is 1/3. More than 4 means 5 or 6, so two outcomes out of six. Note it does not include the 4 itself.

Worked example

The answers

  • P(red) is 4/10, which is 2/5, or 0.4, or 40%.
  • P(not red) is 0.6, either by counting or by 1 − 0.4.
  • Two tails is 1/4. And more than 4 is 2/6, which is 1/3.
Answer
12/5, 0.4, 40%
20.6
31/4
41/3
Summary

What to take away

  • Ways it can happen, over total outcomes, if they are equally likely.
  • One probability, three notations. Convert before you compare.
  • And when a question says at least, reach for the complement.
Remember
P(event)ways it happens over total outcomes
Only whenthe outcomes are equally likely
One probabilitythree notations; convert to compare
An eventcan be several outcomes at once
"At least"is the signal to use the complement

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