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Year 6 Maths Chance

Sum of probabilities of outcomes = 1

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Theory

In any chance experiment, something must happen, so the probabilities of all the possible outcomes add to 1 (or \(100\%\)). This Year 6 maths page shows how to use this to find a missing probability, the chance that an event does not happen, and whether a list of probabilities is possible.

The outcomes of an experiment cover everything that can happen, so their probabilities add to \(1\).

The probability that an event does not happen is \(1\) minus the probability that it does. If the chance of rain is \(0.28\), the chance of no rain is \(0.72\).

With percentages, the total is \(100\%\). With fractions, the total is one whole, such as \(\dfrac{8}{8}\).

A probability can never be less than \(0\) or more than \(1\), so \(1.2\) and \(\dfrac{3}{2}\) cannot be probabilities.

Spinner with three A, two B and one C A spinner with six equal sectors: three labelled A, two labelled B and one labelled C. A A A B B C
\(P(A) = \dfrac{3}{6}\), \(P(B) = \dfrac{2}{6}\), \(P(C) = \dfrac{1}{6}\): together \(\dfrac{6}{6} = 1\).
Probabilities of three colours as a bar A bar standing for a total probability of 1, split into red 0.45, blue 0.3 and an unknown green part. total = 1 red 0.45 blue 0.3 ?
The missing part is \(1 - 0.45 - 0.3 = 0.25\).
Rain and no rain on the scale A probability scale from 0 to 1. Rain is marked at 0.28 and no rain at 0.72; the two points are the same distance from each end. 0 1/4 1/2 3/4 1 rain no rain
\(0.28 + 0.72 = 1\): the two points mirror each other.

Adding to one:

\[ P(\text{all outcomes}) = 1 = 100\% \]
P ( all outcomes )=1=100
\[ P(\text{not } A) = 1 - P(A) \]
P ( not A )=1− P ( A )
\[ \text{missing} = 1 - \text{sum of the others} \]
missing =1− sum of the others
Check the total. If a list adds to more than \(1\) (or \(100\%\)), it is not possible.
Same form. Write all the probabilities as decimals, fractions or percentages before adding.

How to find a missing probability

  1. Add the probabilities you know.
  2. Subtract the total from \(1\) (or from \(100\%\)).
  3. Check that the answer is between \(0\) and \(1\).

How to find the probability of "not"

  1. Find the probability that the event happens.
  2. Subtract it from \(1\).
Example 1 — Not, as a decimal
The chance of rain tomorrow is \(0.28\). What is the chance of no rain?
Solution
\(1 - 0.28\)\(=\)\(0.72\)
1−0.28=0.72

\(0.72\).

Example 2 — Not, as a fraction
The probability of winning a game is \(\dfrac{3}{8}\). What is the probability of not winning?
Solution
\(1 - \dfrac{3}{8}\)\(=\)\(\dfrac{5}{8}\)
1−3/8=5/8

\(\dfrac{5}{8}\).

Example 3 — Not, as a percentage
The chance that a bus is late is \(12\%\). What is the chance it is not late?
Solution
\(100 - 12\)\(=\)\(88\)
100−12=88

\(88\%\).

Example 4 — Three outcomes
A bag holds red, blue and green counters. \(P(\text{red}) = 0.45\) and \(P(\text{blue}) = 0.3\). Find \(P(\text{green})\).
Solution
\(1 - 0.45 - 0.3\)\(=\)\(0.25\)
1−0.45−0.3=0.25

\(0.25\) (see the bar model).

Example 5 — Four outcomes
Three of the four outcomes of a spinner have probabilities \(0.1\), \(0.35\) and \(0.15\). What is the fourth?
Solution
\(0.1 + 0.35 + 0.15\)\(=\)\(0.6\)
\(1 - 0.6\)\(=\)\(0.4\)
1−0.6=0.4

\(0.4\).

Example 6 — Spinner fractions
Use the spinner with A, B and C.
(i) Check that the probabilities add to \(1\).
(ii) Find \(P(\text{not } B)\).
Solution
\(\dfrac{3}{6} + \dfrac{2}{6} + \dfrac{1}{6}\)\(=\)\(1\)
\(1 - \dfrac{2}{6}\)\(=\)\(\dfrac{4}{6}\)
3/6+2/6+1/6=11−2/6=4/6

(i) They add to \(1\).
(ii) \(\dfrac{4}{6} = \dfrac{2}{3}\)

Example 7 — Possible or not?
Could each list be the probabilities of three colours on a spinner?
(i) \(0.55\), \(0.3\), \(0.15\)
(ii) \(0.3\), \(0.3\), \(0.3\)
(iii) \(0.7\), \(1.2\), \(0.1\)
Solution

(i) Yes, they add to \(1\).
(ii) No, they add to \(0.9\).
(iii) No: \(1.2\) is more than \(1\).

Example 8 — Prize wheel
A prize wheel gives a sticker (\(50\%\)), a pencil (\(32\%\)) or a badge.
(i) What is the chance of a badge?
(ii) What is the chance of a sticker or a badge?
Solution
\(100 - 50 - 32\)\(=\)\(18\)
\(50 + 18\)\(=\)\(68\)
1868

(i) \(18\%\)
(ii) \(68\%\)

Example 9 — Find the mistake
Noah lists the chances of three colours as \(0.45\), \(0.35\) and \(0.3\). The first two are right. What is wrong, and what should the third be?
Solution
\(0.45 + 0.35 + 0.3\)\(=\)\(1.1\)
\(1 - 0.45 - 0.35\)\(=\)\(0.2\)
1−0.45−0.35=0.2

The list adds to \(1.1\), but it must add to \(1\). The third should be \(0.2\).

Common pitfalls

Forgetting an outcome. Every possible outcome must be counted in the total of 1.
Mixing forms. Do not add \(0.3\) and \(45\%\) directly: change one first.
Subtracting from 10 or 100 with decimals. With decimals the total is \(1\), not \(10\) or \(100\).
Accepting impossible values. No probability is above \(1\) or below \(0\).

Frequently asked questions

What do all the probabilities of an experiment add to?

\(1\), or \(100\%\).

How do I find the probability that something does not happen?

Subtract the probability that it does happen from 1.

How do I find a missing probability?

Add the others and subtract the total from 1.

Can a probability be 1.2?

No. Probabilities are from 0 to 1.

If the chance of rain is 28%, what is the chance of no rain?

\(72\%\).

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