For equally likely outcomes, \(\text{P(event)} = \dfrac{\text{number of outcomes in the event}}{\text{total number of outcomes}}\). Probabilities lie from 0 (impossible) to 1 (certain), and \(\text{P(not A)} = 1 - \text{P(A)}\).
Fig. 1: the probability scale; picking a red counter from a bag of 3 red and 7 other counters has probability 0.3, so it is unlikely
Section 01
The probability scale.
In the lab, choose how many counters of each colour go in the bag, then pick an event. Watch its marker slide along the scale: make it impossible (0), certain (1), and an even chance (0.5).
0 means impossible and 1 means certain. Nothing is less than 0 or more than 1.
Write a probability as a fraction, decimal or percentage: \(\tfrac{3}{10}\), 0.3 or 30%.
Never write "3 out of 10" or "3 : 10"; these are not accepted.
Equally likely outcomes (a fair dice, a counter picked at random) let you count: wanted outcomes over all outcomes.
Worked example Counters in a bag
A bag contains 3 red, 5 blue and 2 green counters. One counter is taken at random. Find the probability that it is (a) red (b) not green (c) yellow.
Total number of counters \(= 3 + 5 + 2 = 10\), all equally likely.
(a) 3 are red: \(\text{P(red)} = \dfrac{3}{10}\).
(b) \(3 + 5 = 8\) are not green: \(\dfrac{8}{10} = \dfrac{4}{5}\).
(c) There are no yellow counters, so it is impossible: 0.
A probability bigger than 1 or below 0 is always wrong. If you get one, you have probably divided the wrong way round.
Section 02
When it does not happen.
An event either happens or it does not, so the two probabilities add up to 1. Extended candidates use the notation \(\text{P}(A)\) for the probability of \(A\) and \(\text{P}(A')\) for the probability of not \(A\).
Worked example Not happening
The probability that a train is late is 0.12. Find the probability that it is not late.
Late and not late are the only possibilities, so the probabilities add to 1.
\(1 - 0.12 = 0.88\).
Answer: 0.88
Worked example Notation (Extended)
\(\text{P}(B) = 0.8\). Find \(\text{P}(B')\).
\(B'\) means "not \(B\)".
\(\text{P}(B') = 1 - 0.8 = 0.2\).
Answer: \(\text{P}(B') = 0.2\)
Worked example A missing probability
A biased spinner lands on red, blue, green or yellow. P(red) = 0.2, P(blue) = 0.35, and green and yellow are equally likely. Find P(green), then the probability that it does not land on blue.
All four probabilities add to 1: \(0.2 + 0.35 + x + x = 1\).
\(2x = 0.45\), so \(x = 0.225\).
P(not blue) \(= 1 - 0.35 = 0.65\).
Answer: P(green) = 0.225, P(not blue) = 0.65
Your turn
The probability that it rains tomorrow is 37%. What is the probability that it does not rain?
\(100\% - 37\% = 63\%\) (or 0.63).
Examiner trap
The rule \(1 - p\) only works for an event and its opposite. P(not blue) is \(1 - \text{P(blue)}\); it is not P(red), and it is not \(1 - \text{P(red)}\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Which of these cannot be a probability?
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\(\text{P}(A) = 0.35\). What is the probability that \(A\) does not happen?
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A fair six-sided dice is rolled. What is the probability of a score greater than 4?
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Which word best describes a probability of 0.5?
Section 03
Reading tables and Venn diagrams.
Often the outcomes are people or objects counted in a table or a Venn diagram. The probability of picking one at random is still the count you want divided by the total.
Glasses
No glasses
Total
Boys
5
9
14
Girls
7
11
18
Total
12
20
32
Worked example Two-way table
A student is chosen at random from the class in the table. Find the probability that the student is (a) a girl who wears glasses (b) someone who wears glasses.
The total is 32 students.
(a) Girls with glasses: 7, so \(\dfrac{7}{32}\).
(b) Everyone with glasses: \(5 + 7 = 12\), so \(\dfrac{12}{32} = \dfrac{3}{8}\).
In a Venn diagram of 40 people, 15 like tea only, 8 like tea and coffee, 10 like coffee only and 7 like neither. Find the probability that a person chosen at random (a) likes coffee (b) likes neither.
(a) Coffee is the whole coffee circle: \(8 + 10 = 18\), so \(\dfrac{18}{40} = \dfrac{9}{20}\).
(b) Neither is the region outside both circles: \(\dfrac{7}{40}\).
In a Venn diagram, "likes coffee" includes the overlap. Using only the "coffee only" region (10) is the usual slip.
Stage 2 · Exam
Exam-style questions
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A bag contains only red, blue and green counters. P(red) = 0.3 and P(blue) = 0.45. There are 12 green counters. How many counters are in the bag?
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A biased spinner has P(red) = 0.15 and P(blue) = 0.25. The only other colours are green and yellow, and P(green) is twice P(yellow). Find P(green).
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Of 50 students, 23 study French, 19 study Spanish and 8 study both. A student is chosen at random. Find the probability that the student studies neither language.
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A card is chosen at random from cards numbered 1 to 20. Find the probability that the number is prime.
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