Probability Fundamentals

Master key definitions, sample spaces, two-way tables, histograms, and calculating probabilities

SPEC 3.1 • PROBABILITY

Key Definitions

Experiment

A repeatable process that produces well-defined outcomes.

Example: Rolling a dice, flipping a coin, selecting a card

Outcome

A single possible result of an experiment.

Example: Rolling a 4, getting heads, drawing the Ace of Spades

Event

A set of one or more outcomes. Usually denoted by capital letters (A, B, C...).

Example: "Rolling an even number" = {2, 4, 6}

Sample Space (S or ξ)

The set of ALL possible outcomes of an experiment.

Example: For a dice, S = {1, 2, 3, 4, 5, 6}

Remember

A sample space must be mutually exclusive (no overlap) and exhaustive (covers all possibilities).

Key Formulas

Basic Probability

For equally likely outcomes:

P (A) = \frac{n ( A )}{n ( S )}

n(A) = number of favourable outcomes

n(S) = total number of outcomes

Complement Rule

The probability of "not A":

P (A^{'}) = 1 - P (A)

Use when it's easier to count what you don't want.

"At least one" → Find P(none) and subtract from 1

Probability Scale

0 ≤ P(A) ≤ 1 | P(impossible) = 0 | P(certain) = 1 | P(A) + P(A') = 1

Listing Sample Spaces

For simple experiments, we can list all outcomes in the sample space:

ExperimentSample Spacen(S)

Flip a coinS = {H, T}2

Roll a diceS = {1, 2, 3, 4, 5, 6}6

Pick a card suitS = {♠, ♥, ♦, ♣}4

Gender of a babyS = {Boy, Girl}2

Combined Experiments

For n coins: n(S) = 2ⁿ | For two dice: n(S) = 6 × 6 = 36

Sample Space Diagrams

For combined experiments (two events happening together), we use a sample space diagram:

Two Dice: Sum of Scores

Total outcomes: 36 | Highlighted: sum = 7 (6 outcomes)

Reading the Diagram

Each cell represents one outcome. P(sum = 7) = 6/36 = 1/6

Worked Example01

Two fair dice are rolled. Using a sample space diagram, find:

(a) P(sum = 9) (b) P(sum ≤ 4) (c) P(both dice show the same number)

1

Total outcomes in sample space: n(S) = 6 × 6 = 36

a)

Sum = 9 occurs at: (3,6), (4,5), (5,4), (6,3) → 4 outcomes

P (sum = 9) = \frac{4}{36} = \frac{1}{9}

b)

Sum ≤ 4 means sum = 2, 3, or 4

Sum = 2: (1,1) → 1 outcome

Sum = 3: (1,2), (2,1) → 2 outcomes

Sum = 4: (1,3), (2,2), (3,1) → 3 outcomes

Total: 6 outcomes →

P (sum \leq 4) = \frac{6}{36} = \frac{1}{6}

c)

Same number (doubles): (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) → 6 outcomes

P (doubles) = \frac{6}{36} = \frac{1}{6}

Worked Example02

A fair coin is flipped three times. List the sample space and find:

(a) P(exactly 2 heads) (b) P(at least 1 tail)

1

Sample space S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

n(S) = 2³ = 8 outcomes

a)

Exactly 2 heads: {HHT, HTH, THH} → 3 outcomes

P (exactly 2 heads) = \frac{3}{8}

b)

At least 1 tail means "not all heads"

Only outcome with no tails: {HHH} → 1 outcome

P (at least 1 tail) = 1 - P (no tails) = 1 - \frac{1}{8} = \frac{7}{8}

Answers: (a) 3/8 (b) 7/8

Worked Example03

The probability that it rains on any given day in April is 0.35. Find the probability that it does not rain on a randomly chosen day in April.

1

Let R = "it rains". We're given P(R) = 0.35

2

We want P(R') = P(does not rain)

3

Using the complement rule:

P(R') = 1 − P(R) = 1 − 0.35 = 0.65

Answer: P(no rain) = 0.65 or 65%

Two-Way Tables

A two-way table displays data classified by two categories. We can read probabilities directly from it.

MaleFemaleTotal

Left-handed81220

Right-handed423880

Total5050100

Reading Probabilities

P(Male) = 50/100 = 0.5

P(Left-handed) = 20/100 = 0.2

P(Male AND Left-handed) = 8/100 = 0.08 (single cell)

P(Female OR Right-handed) = (50 + 80 − 38)/100 = 0.92 (subtract overlap)

04Worked Example

The table shows data about 120 students and their transport to school.

BusCarWalk

Year 12252015

Year 13183012

Find: (a) P(Year 12) (b) P(Car) (c) P(Year 13 and Bus) (d) P(not Walk)

a)

Year 12 total = 25 + 20 + 15 = 60

P (Year 12) = \frac{60}{120} = \frac{1}{2}

b)

Car total = 20 + 30 = 50

P (Car) = \frac{50}{120} = \frac{5}{12}

c)

Year 13 and Bus = 18

P (Year 13 and Bus) = \frac{18}{120} = \frac{3}{20}

d)

Walk total = 15 + 12 = 27

P (not Walk) = 1 - \frac{27}{120} = \frac{93}{120} = \frac{31}{40}

Worked Example05

In a group of 80 people: 45 are adults, 35 are children. 52 prefer tea, and the rest prefer coffee. 28 adults prefer tea. Construct a two-way table and find P(child who prefers coffee).

1

Fill in what we know: Adults = 45, Children = 35, Tea = 52, Coffee = 80 − 52 = 28

2

Adults who prefer tea = 28, so adults who prefer coffee = 45 − 28 = 17

3

Children who prefer tea = 52 − 28 = 24

4

Children who prefer coffee = 35 − 24 = 11

5

TeaCoffeeTotalAdult281745Child241135Total522880

P(child who prefers coffee) = 11/80

Probability from Continuous Data

For continuous random variables (measurements like time, length, weight), probability is found using proportions or areas.

Uniform Distribution

For a continuous uniform distribution over interval [a, b]:

P (x_{1} \leq X \leq x_{2}) = \frac{x _{2} - x _{1}}{b - a}

Key Insight

Probability = Shaded area ÷ Total area

For uniform distributions: Probability = Length of interval ÷ Total length

06Worked Example

A bus arrives at a stop at a random time between 9:00 and 9:20. Find the probability that the bus arrives:

(a) before 9:08 (b) between 9:05 and 9:15 (c) after 9:12

1

Total interval = 20 minutes (from 9:00 to 9:20)

This is a continuous uniform distribution.

a)

Before 9:08 = first 8 minutes

P (before 9:08) = \frac{8}{20} = \frac{2}{5} = 0.4

b)

Between 9:05 and 9:15 = 10 minutes

P (9:05 to 9:15) = \frac{10}{20} = \frac{1}{2} = 0.5

c)

After 9:12 = last 8 minutes

P (after 9:12) = \frac{8}{20} = \frac{2}{5} = 0.4

Answers: (a) 0.4 (b) 0.5 (c) 0.4

Probability from Histograms

In a histogram, the area of each bar represents frequency (or probability). To find probabilities:

Key Formula

P (value in interval) = \frac{Area of bar(s)}{Total area of histogram}

Remember

Area = Frequency Density × Class Width

If total area = 1, each bar's area directly gives the probability.

If given frequencies, use: P = Frequency in interval ÷ Total frequency

07Worked Example

The histogram shows the time (in minutes) taken by students to complete a task. Find: (a) P(time < 10 mins) (b) P(time ≥ 15 mins)

1

Calculate frequencies using Area = FD × Class Width:

0-5: 2 × 5 = 10 | 5-10: 6 × 5 = 30 | 10-15: 8 × 5 = 40

15-20: 4 × 5 = 20 | 20-30: 1 × 10 = 10

Total frequency = 10 + 30 + 40 + 20 + 10 = 110

a)

Time < 10 mins = classes 0-5 and 5-10

Frequency = 10 + 30 = 40

P (time < 10) = \frac{40}{110} = \frac{4}{11}

b)

Time ≥ 15 mins = classes 15-20 and 20-30

Frequency = 20 + 10 = 30

P (time \geq 15) = \frac{30}{110} = \frac{3}{11}

Answers: (a) 4/11 (b) 3/11

Practice

Question 1: Two fair tetrahedral dice (numbered 1-4) are rolled and the scores are added together. Draw a sample space diagram and find:

(a) P(sum = 5) (b) P(sum is even) (c) P(sum > 6)

Sample space has 4 × 4 = 16 outcomes:

(1,1)=2, (1,2)=3, (1,3)=4, (1,4)=5

(2,1)=3, (2,2)=4, (2,3)=5, (2,4)=6

(3,1)=4, (3,2)=5, (3,3)=6, (3,4)=7

(4,1)=5, (4,2)=6, (4,3)=7, (4,4)=8

(a) Sum = 5: (1,4), (2,3), (3,2), (4,1) → 4 outcomes. P(sum = 5) = 4/16 = 1/4

(b) Even sums (2,4,6,8): 1+4+6+1 = 8 outcomes. P(even) = 8/16 = 1/2

(c) Sum > 6 means 7 or 8: (3,4), (4,3), (4,4) → 3 outcomes. P(sum > 6) = 3/16

Practice

Question 2:

BrownBlondeBlack

Male321825

Female282720

(a) Total Blonde = 18 + 27 = 45. P(Blonde) = 45/150 = 3/10 = 0.3

(b) Male with Black hair = 25. P(Male ∩ Black) = 25/150 = 1/6

(c) Female = 28 + 27 + 20 = 75, Brown = 32 + 28 = 60, Female AND Brown = 28

P(Female or Brown) = (75 + 60 − 28)/150 = 107/150 ≈ 0.713

Practice

Question 3:

Height (cm)0-1010-2020-4040-50

Freq. Density1.5321

Step 1: Calculate areas (frequencies):

0-10: 1.5 × 10 = 15 | 10-20: 3 × 10 = 30 | 20-40: 2 × 20 = 40 | 40-50: 1 × 10 = 10

Total area = 15 + 30 + 40 + 10 = 95

(a) Height < 20 cm: Area = 15 + 30 = 45

P(height < 20) = 45/95 = 9/19 ≈ 0.474

(b) Height ≥ 20 cm: Area = 40 + 10 = 50

P(height ≥ 20) = 50/95 = 10/19 ≈ 0.526

Check: 9/19 + 10/19 = 19/19 = 1 ✓

Summary

Key Definitions & Formulas

ConceptKey Point

Sample Space (S)Set of ALL possible outcomes

Basic ProbabilityP(A) = n(A) / n(S)

ComplementP(A') = 1 − P(A)

Two-Way TablesP = Cell(s) ÷ Total

Continuous/HistogramsP = Area ÷ Total Area

Key Exam Skills

Draw sample space diagrams for combined experiments

Use complement rule for "at least one" questions

Calculate row/column totals in two-way tables

Find areas in histograms using FD × Width