SPI Mathematics · Probability

Probability Rules and the Complement

A probability always lies from 0 to 1, and the chance that an event does not happen is 1 minus the chance that it does.

Why this matters

Eleven questions in the bank, all EASY and all one line: which value cannot be a probability, and P(not E) from P(E). The printed wrong options are subtraction slips such as 53% for 1 minus 57%, so check the arithmetic, not the idea.

Concept 1 of 2: Every probability lies from 0 to 1

The favourable outcomes can never outnumber all the outcomes, and there can never be fewer than none, so the fraction is never below 0 or above 1.

Definition

An impossible event has probability 0: a 7 on a die, a sum of 1 with two dice.
A sure event has probability 1: a number from 1 to 6 on a die.
Every other event lies strictly between: $0 < P(E) < 1$ .
As a percentage the range is 0% to 100%. A decimal such as 0.05 or a percentage such as 5% is allowed; a negative value or anything above 1 is not.

Range of a probability

0 \le P(E) \le 1

Worked example

Which of 0.7, 1.05,

\frac{3}{4}

and 0% cannot be the probability of an event?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Probability · Probability Rules and the Complement

Which of the following can't be the probability of an event?

Reading a percentage as a whole number

5% is 0.05, a valid probability. As a percentage a probability can be any value from 0 to 100, including decimals such as 12.5%.

Allowing negative probabilities

-1 \le P(E) \le 1

is a printed wrong option. A probability is never negative, so the range starts at 0.

Drill 4 more on every probability lies from 0 to 1

Concept 2 of 2: The probability of an event not happening

Every outcome either belongs to the event or does not, so the two chances share the whole of 1 between them. Find one and the other is what is left.

Definition

'Not E' is the complement of E: every outcome that is not in E.
$P(E) + P(\text{not } E) = 1$ , so $P(\text{not } E) = 1 - P(E)$ .
Losing is the complement of winning; failure is the complement of success.
Work in one form throughout: $1 - 0.57 = 0.43$ , or $100\% - 57\% = 43\%$ .

Complement rule

P(\text{not } E) = 1 - P(E)

Worked example

The probability of an event is 0.36. What is the probability that it does not happen?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Probability · Probability Rules and the Complement

If probability of happening of an event is

57\%

, then probability of non - happening of the event is

Subtracting from 100 in the wrong column

100% minus 57% is 43%, not 53%. Write the subtraction down: a slip in the tens digit is exactly what the wrong options print.

Answering with the event instead of its complement

When P(E) = 0.57 is given, 0.57 is often among the options. The question asks for 'not E', so the answer is the other part of 1.

Drill 5 more on the probability of an event not happening

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Every probability lies from 0 to 1
Range of a probability
$0 \le P(E) \le 1$
The probability of an event not happening
Complement rule
$P(\text{not } E) = 1 - P(E)$

Watch out for (4)

Reading a percentage as a whole number
→ Every probability lies from 0 to 1
Allowing negative probabilities
→ Every probability lies from 0 to 1
Subtracting from 100 in the wrong column
→ The probability of an event not happening
Answering with the event instead of its complement
→ The probability of an event not happening

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