1. Definition of Probability
Probability of an event is defined as: where:
- is the probability of event .
- Favorable outcomes are the outcomes that satisfy the event .
- Total outcomes are the total number of possible outcomes in the experiment.
2. Sample Space (S)
The set of all possible outcomes of a random experiment is called the sample space. For example:
- Tossing a coin:
- Rolling a die:
3. Event
An event is a subset of the sample space. For example, when rolling a die, getting an even number () is an event.
Important Points to Remember
Probability always lies between 0 and 1, inclusive: .
Probability of an impossible event is 0.
Probability of a sure event is 1.
Probability : Solved Examples
Example 1: Tossing a Coin
A coin is tossed once. What is the probability of getting a head?
Solution:
Sample space:
Total outcomes = 2
Favorable outcomes (Head) = 1
Example 2: Rolling a Die
A die is rolled once. What is the probability of getting a number greater than 4?
Solution:
Sample space:
Total outcomes = 6
Favorable outcomes ({5, 6}) = 2
Example 3: Drawing a Card
A card is drawn from a standard deck of 52 cards. What is the probability of drawing a king?
Solution:
Total outcomes = 52
Favorable outcomes (Kings) = 4
Structure of a Standard Deck
- Total Cards: 52 cards
- Suits: 4 suits (Hearts, Diamonds, Clubs, Spades) with 13 cards in each suit.
- Hearts and Diamonds are red.
- Clubs and Spades are black.
- Ranks: Each suit contains:
- Number cards: 2 through 10
- Face cards: Jack, Queen, King
- Ace
Applications in Probability
Basic Probability Concepts
- Single Event Probability: Calculating the chance of drawing a specific card, such as an Ace or a Heart. P(Event)=Number of favorable outcomes/Total outcomes
- Example: Probability of drawing an Ace = 4/52=1/13
Compound Events
- Independent Events: Drawing a card, replacing it, and drawing again.
- Dependent Events: Drawing a card, not replacing it, and drawing again. Example: Probability of drawing two Aces without replacement:
Complementary Events
- Probability of an event not happening. Example: Probability of not drawing a Heart = 1−13/52=39/52