7 min readConcept sheetPart I MCQ and Part II (2 marks); tree diagrams often help in Part III

Random experiments, sample space and events

A random experiment has known outcomes but an unpredictable result. The sample space S lists every outcome, and an event is any subset of S.

Random experimentAll outcomes are known, but the result of one trial cannot be predicted

Sample space SThe set of all possible outcomes

EventAny subset of the sample space

Number of outcomesn coins: 2ⁿ; n dice: 6ⁿ

Experiments and sample space

An experiment is an action with well-defined results. In a deterministic experiment the result is known in advance: heating water to 100 °C at sea level makes it boil. In a random experiment we know all the possible results but cannot say which one will happen in a single try: tossing a coin, rolling a die. Each possible result is an outcome or sample point. The set of all outcomes is the sample space S. Any subset of S is an event.

Types of events

Type	Meaning	Example (one die rolled)
Equally likely events	Each has the same chance of happening	Getting 1, 2, 3, 4, 5 or 6 on a fair die
Certain (sure) event	Always happens; it is S itself	Getting a number less than 7
Impossible event	Never happens; it is the empty set ∅	Getting 8
Mutually exclusive events	Cannot happen together: A ∩ B = ∅	Getting an odd number and getting an even number
Exhaustive events	Together they cover all of S	{1, 2, 3} and {3, 4, 5, 6}
Complementary events	Mutually exclusive and exhaustive: A and Ā	{1, 2} and {3, 4, 5, 6}
Simple (elementary) event	Has only one outcome	Getting 5

Sample spaces to know

One coin: {H, T}. Two coins: {HH, HT, TH, TT}. Three coins: 8 outcomes.
One die: {1, 2, 3, 4, 5, 6}. Two dice: 6 × 6 = 36 outcomes, from (1, 1), (1, 2) up to (6, 6).
n coins give 2ⁿ outcomes; n dice give 6ⁿ outcomes.
A pack of 52 cards has 4 suits of 13 cards: spades and clubs (black), hearts and diamonds (red). Each suit has A, 2 to 10, J, Q and K. The 12 face cards are the J, Q and K of each suit.

How to draw a tree diagram

Example: a coin is tossed and then a die is rolled.

1
Draw a dot on the left for the start.
2
Draw one branch for each outcome of the first action and label them H and T.
3
From the end of each branch, draw six branches for the die and label them 1 to 6.
4
Read each path from left to right to write an outcome: H1, H2, and so on up to T6.
5
Count the paths: 2 × 6 = 12 outcomes, so n(S) = 12.

Example: three coins

Problem

Three coins are tossed together. Write the sample space and the event A: getting at least two heads.

Working

Each coin gives H or T, so there are 2 × 2 × 2 = 8 outcomes.
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}, n(S) = 8
At least two heads means two heads or three heads.
A = {HHH, HHT, HTH, THH}, n(A) = 4

Answer

n(S) = 8; A = {HHH, HHT, HTH, THH}

In real life

Every throw of the dice in Ludo or Paramapadham (snakes and ladders) is a random experiment with sample space {1, 2, 3, 4, 5, 6}.

Exam angle

1 marks: “How many outcomes are there when three coins are tossed?” 2³ = 8.

2 marks: “A coin is tossed and a die is rolled together. Write the sample space.” S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}, so n(S) = 12.

5 marks: “A coin is tossed and a die is rolled together. Draw a tree diagram and write the sample space. List the events A: a head with an even number, and B: a tail with a number greater than 4. Are A and B mutually exclusive?” Tree with branches H, T, then 1 to 6. S has 12 outcomes. A = {H2, H4, H6}, B = {T5, T6}. A ∩ B = ∅, so A and B are mutually exclusive.

Remember it

S holds every outcome; an event is a piece of S. Exclusive means no overlap; exhaustive means nothing left out.