Introduction to probability IGCSE

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📊 Introduction to Probability - Complete Theory & Formulas

What is Probability?
Probability measures how likely an event is to happen. It is always a number between 0 and 1, where 0 = impossible and 1 = certain.

🔑 Key Terms (Essential Vocabulary)

Experiment: An action with an uncertain outcome (e.g., tossing a coin, rolling a die)

Outcome: One possible result of an experiment (e.g., "Head" or "4")

Event: A set of outcomes (e.g., "getting an even number")

Sample Space: The complete list of all possible outcomes, written as {1,2,3,4,5,6}

Favourable Outcomes: Outcomes that match the event you want

Frequency: How many times an outcome appears in trials

Probability Scale (0 to 1)

📚 Section 8.1: Basic Probability (Experimental)

Experimental Probability Formula:

P(A) = Number of times A happens ÷ Total number of trials

Example: Drawing counters from a bag 100 times

Colour	Frequency	Probability
Red	44	44/100 = 0.44
White	32	32/100 = 0.32
Green	24	24/100 = 0.24

Total = 100 trials

Key Idea: The more trials you do, the closer experimental probability gets to theoretical probability.

📐 Section 8.2: Theoretical Probability

Theoretical Probability Formula:

P(event) = Number of favourable outcomes ÷ Number of possible outcomes

When to use: When all outcomes are equally likely (fair coin, fair die, etc.)

Example 1: Rolling a fair die
Sample space: {1, 2, 3, 4, 5, 6}
Total possible outcomes = 6

P(even number) = 3/6 = 1/2
Favourable outcomes: {2, 4, 6}

P(number > 4) = 2/6 = 1/3
Favourable outcomes: {5, 6}

Example 2: Tossing a fair coin
Sample space: {H, T}
P(H) = 1/2
P(T) = 1/2

Important: All probabilities in a complete sample space add up to 1.
P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1

🔄 Section 8.3: Complement (Event NOT Happening)

Complement Rule:

P(not A) = 1 - P(A)

Example: Ice cream flavours

Flavour	Probability
Strawberry	0.25
Lime	0.15
Lemon	0.20
Blackberry	0.18
Apple	0.22

P(not Strawberry) = 1 - 0.25 = 0.75

Quick Check: To find missing probability when others are given:
Missing probability = 1 - (sum of given probabilities)

📋 Section 8.4: Possibility Diagrams

What is it? A table showing ALL combined outcomes for two events (e.g., rolling two dice, choosing drink + snack).

Total Outcomes Formula:

Total = (Outcomes from Event 1) × (Outcomes from Event 2)

Example: Rolling two dice

Die 2 → Die 1 1 2 3 4 5 6 ↓ +---+---+---+---+---+---+ 1 |1,1|1,2|1,3|1,4|1,5|1,6| +---+---+---+---+---+---+ 2 |2,1|2,2|2,3|2,4|2,5|2,6| +---+---+---+---+---+---+ 3 |3,1|3,2|3,3|3,4|3,5|3,6| +---+---+---+---+---+---+ 4 |4,1|4,2|4,3|4,4|4,5|4,6| +---+---+---+---+---+---+ 5 |5,1|5,2|5,3|5,4|5,5|5,6| +---+---+---+---+---+---+ 6 |6,1|6,2|6,3|6,4|6,5|6,6| +---+---+---+---+---+---+

Total outcomes = 6 × 6 = 36

P(sum = 7) = 6/36 = 1/6
Favourable: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)

Example: Drinks and Snacks
Drinks: Water (W), Juice (J), Cola (C)
Snacks: Crisps (Cr), Sandwich (S), Fruit (F)

	Crisps	Sandwich	Fruit
Water	W,Cr	W,S	W,F
Juice	J,Cr	J,S	J,F
Cola	C,Cr	C,S	C,F

Total = 3 × 3 = 9 outcomes
P(Juice and Fruit) = 1/9
P(any drink with Fruit) = 3/9 = 1/3

🔗 Section 8.5: Independent and Mutually Exclusive Events

Independent Events

Definition: Two events are independent if one event does NOT affect the other.
Examples: Tossing a coin AND rolling a die; rolling two separate dice.

Independent Events Rule:

P(A and B) = P(A) × P(B)

Example: Toss a coin AND roll a die
P(Head) = 1/2
P(6) = 1/6

P(Head AND 6) = 1/2 × 1/6 = 1/12

Mutually Exclusive Events

Definition: Two events are mutually exclusive if they CANNOT both happen at the same time.
Examples: Getting Head AND Tail in one toss; rolling 3 AND 5 on one die.

Mutually Exclusive Events Rule:

P(A or B) = P(A) + P(B)
(when they cannot both happen)

Example: Rolling a die
P(3) = 1/6
P(5) = 1/6

P(3 OR 5) = 1/6 + 1/6 = 2/6 = 1/3
(These are mutually exclusive - you can't roll 3 AND 5 in one roll)

When events CAN both happen (overlap):

P(A or B) = P(A) + P(B) - P(A and B)

Example with overlap: Rolling a die
A = even number {2, 4, 6}
B = number > 4 {5, 6}

P(A) = 3/6 = 1/2
P(B) = 2/6 = 1/3
P(A and B) = 1/6 (the 6 is in both)

P(A or B) = 1/2 + 1/3 - 1/6 = 3/6 + 2/6 - 1/6 = 4/6 = 2/3

🎯 Complete Formula Summary

1. Experimental Probability
P(A) = Frequency of A ÷ Total trials

2. Theoretical Probability
P(event) = Favourable outcomes ÷ Possible outcomes

3. Complement Rule
P(not A) = 1 - P(A)

4. Independent Events (AND)
P(A and B) = P(A) × P(B)

5. Mutually Exclusive (OR)
P(A or B) = P(A) + P(B)

6. General OR Rule
P(A or B) = P(A) + P(B) - P(A and B)

7. Total Probability
Sum of all probabilities in sample space = 1

🔍 Problem-Solving Strategy

Step 1: Identify the experiment and write the sample space

Step 2: Count total possible outcomes

Step 3: Count favourable outcomes for your event

Step 4: Decide which rule to use:

• Basic fraction? Use P = favourable ÷ possible
• NOT happening? Use P(not A) = 1 - P(A)
• Two independent events? Use P(A and B) = P(A) × P(B)
• Mutually exclusive OR? Use P(A or B) = P(A) + P(B)
• Two events together? Draw possibility diagram

Step 5: Calculate and simplify

Step 6: Check: Is 0 ≤ answer ≤ 1? ✓

📱 Quick Reference Table

Situation	Formula	When to Use
Single event	P = n/N	Fair outcomes, one choice
NOT event	P = 1 - P(A)	"Does not", "other than"
A AND B	P(A) × P(B)	Independent events, both happen
A OR B	P(A) + P(B)	Mutually exclusive, one happens
Two choices	Draw diagram	Combined outcomes, pairs
From data	frequency ÷ total	Experimental results given

✅ Key Points to Remember

• Probability is ALWAYS between 0 and 1
• Can write as fraction, decimal, or percentage
• All probabilities in a sample space add to 1
• Use × for independent events (AND)
• Use + for mutually exclusive events (OR)
• Use 1 - for complement (NOT)
• Draw diagrams for two-step experiments
• More trials → better estimate of probability
• Sample space = list of ALL possible outcomes
• Always simplify fractions in final answer

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