1 Probability
Probability deals with the chance (likelihood) of the occurrence of an event in a random experiment.
2 Random Experiment
A random experiment is an experiment in which:
- All possible outcomes are known in advance (predictable outcomes set).
- The exact outcome cannot be predicted with certainty (unpredictable exact result).
3 Sample Space (S)
The sample space is the set of all possible outcomes of a random experiment.
Example (Die)
S = {1, 2, 3, 4, 5, 6}
Example (Two Coins)
S = {(H,H), (H,T), (T,H), (T,T)}
4 Event
An event is a subset of the sample space. (It is a set of outcomes for which we say the event “occurs”.)
Example (Odd number on die)
A = {1, 3, 5}
Example (One head and one tail)
A = {(H,T), (T,H)}
5 Occurrence of an Event
If the actual outcome of the experiment is a member of the event set, the event is said to occur; otherwise, it does not occur.
Example (Die)
If A = {2, 4} and the result is 4, then A occurs. If the result is 5, then A does not occur.
6 Elementary (Simple) Event
An elementary event contains exactly one sample point.
Example
Getting 3 in a die throw: {3}
7 Compound (Composite) Event
A compound event contains more than one sample point.
Example (Odd numbers)
{1, 3, 5}
8 Sure Event
A sure event is certain to occur (it contains all outcomes of the sample space).
Example (Die)
Getting a number from 1 to 6: S itself (sure event).
9 Impossible Event
An impossible event cannot occur (it has no outcomes).
Example (Die)
Getting 7 in a single throw: ∅ (impossible event).
10 Equally Likely Outcomes
Outcomes are equally likely if none of the outcomes is expected to occur in preference to the other.
Coin
P(H) = 1/2 and P(T) = 1/2 (fair coin).
Die
Each outcome has probability 1/6 (fair die).
11 Complementary Events
If A is an event, then its complement A′ is the set of all outcomes in S that are not in A.
Example (Die)
A = {1, 3, 5} (odd), then A′ = {2, 4, 6} (even).
12 Exhaustive Events (Exhaustive Outcomes)
The total number of possible outcomes of a random experiment is called the number of exhaustive outcomes.
- Tossing a coin → Exhaustive outcomes =
2 - Throwing a die → Exhaustive outcomes =
6 - Drawing 4 cards from 52 cards → Number of outcomes =
⁵²C₄
13 Mutually Exclusive Events
Two (or more) events are mutually exclusive if they cannot occur simultaneously.
Example (Single card draw)
Event A: “Spade” and Event B: “Heart” cannot occur together → mutually exclusive.
14 Not Mutually Exclusive Events
Events are not mutually exclusive if they can occur simultaneously.
Example (Single card draw)
Event A: “King” and Event B: “Spade” can occur together (King of Spades) → not mutually exclusive.