Probabilities that sum to one
Every probability sits between 0 and 1, and a complete set of outcomes always totals exactly 1. That single fact answers a surprising number of questions.
Lesson overview
What you'll learn in this lesson
Understand that the probabilities of all possible outcomes sum to 1
Key learning points
- • The scale from 0 to 1
- • Using the total
- • The complement
- • Expected frequency
This lesson at a glance
- 30 minutes
- 19 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 19
Watch first
Probabilities that sum to one
An English teaching video on this topic from mathantics. Watch it, then scroll down for the lesson.
In a nutshell
Understand that the probabilities of all possible outcomes sum to 1
What you already know
Part 2 of 19
Before we start
What you already know
You can write probabilities as fractions, decimals and percentages.
Key words
Part 3 of 19
Key words
Words you'll need today
- outcome
- One possible result of a trial.
- mutually exclusive
- Outcomes that cannot both happen at once.
- complement
- Everything except the event; its probability is 1 − P(event).
Learning cycle
Part 4 of 19
Learning cycle 1 of 2
Part 1 — The total is 1
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 5 of 19
Learn
The scale from 0 to 1
Impossible is 0, certain is 1, and everything else lies between. A probability of 1.4 or −0.2 is not a mistake to argue about — it is meaningless. Fractions, decimals and percentages are all acceptable, but never write odds like '3 in 4' as 3/4 unless it genuinely means 3 out of 4.
Explore the idea
Part 6 of 19
Learn
Using the total
A spinner has red 0.3, blue 0.45 and green unknown. Since the three cover every outcome, green = 1 − 0.75 = 0.25. If two unknowns are equal, share the remainder between them. This is the most common structured question in the topic.
Quick check
Part 7 of 19
Quick check
P(A) = 0.62. What is P(not A)?
Quick check
Part 8 of 19
Quick check
Red 0.25, blue 0.4, green x. Find x.
Reset break
Part 9 of 19
Pause
That's sitting 1 of 2 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Learning cycle
Part 10 of 19
Learning cycle 2 of 2
Part 2 — Expected results
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 11 of 19
Learn
The complement
P(not A) = 1 − P(A). If the probability of rain is 0.35, the probability of no rain is 0.65. This is often much quicker than adding up every other case — for two dice, P(not a double) = 1 − 6/36 = 30/36.
Explore the idea
Part 12 of 19
Learn
Expected frequency
Expected frequency = probability × number of trials. With P(red) = 0.2 and 300 spins, expect 60 reds. Getting 54 is not evidence the spinner is biased; small differences are normal. Larger experiments give relative frequencies closer to the true probability.
Quick check
Part 13 of 19
Quick check
Which cannot be a probability?
Quick check
Part 14 of 19
Quick check
P(win) = 0.15. Expected wins in 400 games?
Common mix-ups
Part 15 of 19
Common mix-ups
Things people often get wrong
- Giving a probability greater than 1.
- Forgetting that the listed outcomes must cover everything before subtracting from 1.
- Expecting the exact expected frequency to happen every time.
Challenge round
Part 16 of 19
Game · Sort it
Which of these are true?
Drag each card into the right column. Tap a card first if dragging is fiddly.
True
Not true
Challenge round
Part 17 of 19
Game · Fill the gaps
Finish the sentences
Choose the word that belongs in each gap.
____ frequency = probability × number of trials.
Mastery quiz
Part 18 of 19
Marked quiz
End of lesson quiz: Probabilities that sum to one
6 questions, marked with the reasoning shown. No timer.
1. All probabilities of a complete set sum to...
2. P(late) = 1/8. P(not late)?
3. Two equal unknowns share a remainder of 0.36. Each is...
4. A fair coin, 500 flips. Expected heads?
5. Getting 243 heads in 500 flips suggests...
6. A certain event has probability...
Lesson round-up
Part 19 of 19
Lesson round-up
Ready when you are
Quiz score
Not sat
Games
Not played
Points this lesson
0
Best run
0 in a row
Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?
Ask LunaDo the work
Your written work for this lesson
This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.
The cost of a school trip
A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.
If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.
The school decides to charge £20 and to use anything left over for a second trip later in the year.
Written for Good Learning Co.
Reading questions
- Work out the total cost of the trip if 50 pupils go.
- Show why the price per pupil falls when more pupils go, using numbers.
- At £20 each, how many pupils must go before the trip makes money? Show your working.
Working out
Working outDo the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.
10–15 questions
Worked example, explained
WritingCopy one question out and write underneath, in words, why each step works.
1 question, 4–5 sentences
Problem in words
Working outWrite your own real-life problem that uses today's method, then solve it.
1 problem + solution
Saved as you type.
Part 1 of 19 · Watch & discover