KS3 · Maths

Probabilities of all outcomes sum to 1

On a dice, P(factor of 10) is 3/6 and P(factor of 12) is 5/6. Add them: 8/6. That is over 1, and 1 means certain. What went wrong?

Try it first

Catch the double count

Roll a fair six-sided dice. Take two events: 'factor of 10' and 'factor of 12'. For each possible outcome, tick every event it belongs to. It can be one, both or neither. Then check.

  • A Factor of 10
  • B Factor of 12
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  6. 6

Now test some pairs

Two tests, one total

For each pair of events, tick every test it passes. Be fussy: check outcome by outcome, not by how the events sound. Then check, and look down the last column.

Dice: an even number, an odd number
Spinner 1 to 5 spun twice, totals added: a prime total, a total that is not prime
Dice: a multiple of 3, a multiple of 4
Dice: a factor of 10, a factor of 12

See 'one whole' as a length

Rain and not rain

P(rain) = 0.35. Slide 'rain' to 0.35. Then slide 'not rain' to where you think it belongs. Check to see where they really sit.

Find the missing probability

Problem

Alex and Izzy play one chess match. It can end three ways: Alex wins, Izzy wins or it is a draw. P(Alex wins) = 0.4 and P(Izzy wins) = 0.45. Find P(draw).

Maths · Algebra

When the probabilities are unknown

Three runners race. Exactly one of them wins. Andeep and Sofia are each twice as likely to win as Laura.

GoalFind P(Laura wins)
1
P(Laura) = x, P(Andeep) = 2x, P(Sofia) = 2x
Call Laura's probability x, then write the other two in terms of x.
2
3
4
5

Step 1 of 5

Working step.

Watch out: x is only Laura's probability. Andeep's and Sofia's are 2x, so 0.4 each.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Only when the events cover every outcome exactly once

What you need to know

  • The probabilities of all possible outcomes of a trial add up to 1, which means certain: one of them must happen.
  • Events are mutually exclusive if they share no common outcome, so no outcome can satisfy two of them.
  • Have a goZara says 'roll a 2' and 'roll an even number' are mutually exclusive, because they sound like different things. Is she right?

    No. Rolling a 2 is an outcome that satisfies both events.

    Mutually exclusive means no shared outcome. How different the events sound makes no difference.

  • Events are exhaustive if at least one of them has to happen every time, so no outcome is left out.
  • Have a goOn one roll of a fair dice, take the events 'less than 3' and 'more than 4'. Is that pair exhaustive?

    No. Rolling a 3 or a 4 satisfies neither event.

    Exhaustive means at least one event happens every time, and 3 and 4 slip through the gap.

  • Mutually exclusive and exhaustive events cover every outcome exactly once, so their probabilities sum to 1.
  • If events overlap or leave a gap, their probabilities may not sum to 1: an overlap counts outcomes twice, a gap misses some.
  • A Venn diagram shows both tests: an empty overlap means mutually exclusive, and an empty outside region means exhaustive.
  • The probability that an event does not happen is 1 minus the probability that it does.
  • Have a goThe chance a bus is late is 0.2. What is the chance it is not late?

    0.8

    'Not late' covers every other outcome, so you take the probability of 'late' away from 1: 1 − 0.2 = 0.8.

  • In a mutually exclusive, exhaustive set, a missing probability is 1 minus the sum of the known ones.
  • Probabilities can be fractions, decimals or percentages, and one whole is 100%.
  • If probabilities are unknown but related, write an expression for each, set their sum equal to 1 and solve.

The big picture

The probabilities of all the possible outcomes of a trial add up to 1. A set of events is only certain to add up to 1 when it is mutually exclusive (no shared outcome) and exhaustive (no outcome left out). That one idea lets you find a missing probability, work out P(not A) and solve for related unknown probabilities.

Key points

1All the possible outcomes of a trial have probabilities that add to 1, and 1 means certain.
2Events are certain to sum to 1 only when they are mutually exclusive (no shared outcome) and exhaustive (no outcome left out).
3An overlap counts outcomes twice and a gap misses some, so either can stop the total being 1.
4A missing probability is 1 minus the sum of the known ones, and P(not A) = 1 − P(A).
5For related unknowns, write each as an expression, set the sum equal to 1 and solve.

Worked example

Problem

A bag holds only red, blue and yellow counters. One counter is taken at random. P(red) = 3/10 and P(blue) = 1/2. Find P(yellow).

⚠ Watch out

Assuming the probabilities of any events you list for a trial add up to 1. They only have to when the events are mutually exclusive and exhaustive, so check for an overlap and for a gap before you trust the total.

🧠

Memory hook

No overlap, no gap, total 1. Cover every outcome once and only once.

✓

Check yourself

Odd {1, 3, 5} and prime {2, 3, 5} both have probability 3/6, so the total is 1. Does that make them mutually exclusive and exhaustive? Hint: check 3 and 5, then 4 and 6.

Flashcards

(12)
What do the probabilities of all possible outcomes of a trial add up to?
1. And 1 means certain, because one of the possible outcomes is certain to happen.
When are events mutually exclusive?
When they share no common outcome. No outcome can satisfy two of them.
When is a set of events exhaustive?
When at least one of them has to happen every time. No outcome satisfies none of them.
Which two tests make a set of events certain to sum to 1?
Mutually exclusive and exhaustive. Together they cover every outcome exactly once.
What goes wrong with the total if events are not mutually exclusive?
Some outcomes are counted twice, so the probabilities may not sum to 1.
What goes wrong with the total if events are not exhaustive?
It is not certain that one of them happens, so the probabilities will not sum to 1.
On a Venn diagram of two events, what does an empty overlap tell you?
The events are mutually exclusive: no outcome is in both.
On a Venn diagram of two events, what does an empty region outside both sets tell you?
The events are exhaustive: no outcome is left out.
You know P(A). How do you find the probability that A does not happen?
Take it away from 1: P(not A) = 1 − P(A).
One probability is missing from a mutually exclusive, exhaustive set. How do you find it?
Add the known probabilities, then subtract that total from 1.
In percentages, what is one whole?
100%. So 100% − 70% = 30% does the same job as 1 − 0.7 = 0.3.
Three probabilities are unknown but related to each other. What is your plan?
Write an expression for each, set their sum equal to 1, then solve the equation.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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