1. Meaning of Probability
Probability is a way of measuring how likely an event is to happen. It gives a number between 0 and 1 to describe the chance of an event.
0 means the event is impossible, and 1 means the event is certain. Anything in between shows different levels of chance.
Examples:
- Getting a head when flipping a coin → chance is 1/2.
- Rolling a 7 on a normal die → chance is 0 (impossible).
- The sun rising tomorrow → chance is almost 1 (certain).
2. Experiments, Outcomes and Sample Space
A probability experiment is any action where the result is uncertain. Each possible result is called an outcome.
The list of all possible outcomes is called the sample space and is usually written using curly brackets.
2.1. Example
- Experiment: Tossing a coin
- Outcomes: {H, T}
- Sample space: S = {H, T}
- Experiment: Rolling a die
- Outcomes: {1, 2, 3, 4, 5, 6}
- Sample space: S = {1, 2, 3, 4, 5, 6}
3. Events
An event is a collection of one or more outcomes from the sample space. Events describe something happening.
3.1. Types of Events
- Simple event: Contains only one outcome. Example: Getting a 4 when rolling a die.
- Compound event: Contains more than one outcome. Example: Getting an even number {2, 4, 6}.
- Certain event: Always happens. Example: Getting a number from 1 to 6 when a die is rolled.
- Impossible event: Can never happen. Example: Getting an 8 on a die.
4. Classical (Theoretical) Probability
The classical formula is used when all outcomes are equally likely.
\( P(E) = \dfrac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}} \)
This formula is used for coins, dice, cards and other cases where every outcome has the same chance.
4.1. Example
Finding the probability of getting a 3 when a die is rolled:
- Favourable outcomes = 1
- Total outcomes = 6
\( P(3) = 1/6 \)
5. Important Properties of Probability
Some basic facts about probability that are always true:
- Property 1: Probability of any event lies between 0 and 1.
\( 0 \le P(E) \le 1 \)
- Property 2: Sum of probabilities of all outcomes in the sample space is 1.
- Property 3: Probability of “not E”:
\( P(\text{not } E) = 1 - P(E) \)
5.1. Example
If the probability of raining today is 0.3, then the probability of not raining is:
\( P(\text{not rain}) = 1 - 0.3 = 0.7 \)