Probability that adds to one
Probability is a measure of expectation on a scale from 0 to 1. The single most useful fact is that the probabilities of all possible outcomes must total exactly 1.
Part of your national curriculum
- Probability: Understand that the probabilities of all possible outcomes sum to 1
Lesson overview
What you'll learn in this lesson
Understand that the probabilities of all possible outcomes sum to 1
Key learning points
- • The total is always 1
- • Complementary events
- • Expected frequency
- • Relative frequency and fairness
This lesson at a glance
- 28 minutes
- 16 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 16
Watch first
Probability that adds 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
Learning cycle
Part 2 of 16
Learning cycle 1 of 2
Part 1 · The total is always 1
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 16
Learn
The total is always 1
If a spinner lands on red with probability 0.3 and blue with 0.45, then green must have probability 1 − 0.3 − 0.45 = 0.25. This 'complete the set' reasoning solves a large share of exam questions with one subtraction.
Explore the idea
Part 4 of 16
Learn
Complementary events
The probability that an event does not happen is 1 minus the probability that it does. If the chance of rain is 0.18, the chance of no rain is 0.82. Recognising a question as a complement is often much faster than adding many separate cases.
Quick check
Part 5 of 16
Quick check
P(red) = 0.3, P(blue) = 0.45. P(green) is
Quick check
Part 6 of 16
Quick check
If P(rain) = 0.18, P(no rain) is
Reset break
Part 7 of 16
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 8 of 16
Learning cycle 2 of 2
Part 2 · Expected frequency
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 9 of 16
Learn
Expected frequency
Expected frequency is probability multiplied by the number of trials. With P(six) = 1/6 and 300 rolls, you expect about 50 sixes. 'About' matters: actual results vary, and expected frequency describes a long-run tendency, not a guarantee.
Explore the idea
Part 10 of 16
Learn
Relative frequency and fairness
Relative frequency is the number of successes divided by the number of trials, and it estimates probability better as the number of trials grows. If a die gives 90 sixes in 300 rolls when 50 were expected, that is evidence of bias, though not proof from one experiment alone.
Quick check
Part 11 of 16
Quick check
With P(six) = 1/6, expected sixes in 300 rolls is
Quick check
Part 12 of 16
Quick check
Relative frequency becomes a better estimate when
Explore the idea
Part 13 of 16
Worked example
Worked answer: a spinner has P(red) = 0.3 and P(blue) = 0.45. Find P(green) and the expected greens in 200 spins (3 marks)
All probabilities total 1, so P(green) = 1 − 0.3 − 0.45 = 0.25 (1). Expected frequency = probability × trials = 0.25 × 200 (1) = 50 greens (1). This is an expectation over many spins, so an actual set of 200 spins might give 44 or 57 greens without the spinner being unfair.
Challenge round
Part 14 of 16
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
Mastery quiz
Part 15 of 16
Marked quiz
End of lesson quiz: Probability that adds to one
4 questions, marked with the reasoning shown. No timer.
1. P(red) = 0.3, P(blue) = 0.45. P(green) is
2. If P(rain) = 0.18, P(no rain) is
3. With P(six) = 1/6, expected sixes in 300 rolls is
4. Relative frequency becomes a better estimate when
Lesson round-up
Part 16 of 16
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 16 · Watch & discover
How this is going
- Not looked at yet
Understand that the probabilities of all possible outcomes sum to 1