Concept of Probability

Probability measures the likelihood of an event occurring within a range from 0 to 1:

• 0: The event is impossible.
• 1: The event is certain.
• Between 0 and 1: Indicates varying levels of uncertainty. A higher value denotes greater likelihood.

Key Terms in Probability

1. Experiment:
   • Any process or action that generates a set of outcomes.
   • Example: Flipping a coin, rolling a die.
2. Outcome:
   • A possible result of an experiment.
   • Example: Heads in a coin flip, rolling a 4 on a die.
3. Sample Space (S):
   • The set of all possible outcomes of an experiment.
   • Example: For a coin flip, S={Heads, Tails} 
4. Event (E):
   • A subset of the sample space, representing one or more outcomes of interest.
   • Example: Getting an even number when rolling a die (E={2,4,6} 

Types of Events in Probability

1. Independent Events:
   • Events that do not influence each other.
   • Example: Rolling a die and flipping a coin simultaneously.
2. Dependent Events:
   • Events where the occurrence of one affects the likelihood of the other.
   • Example: Drawing cards without replacement.
3. Mutually Exclusive Events:
   • Events that cannot occur simultaneously.
   • Example: Rolling a 3 or a 5 on a die.
4. Complementary Events:
   • The event and its complement together cover the entire sample space.
   • Example: If EEE is rolling an even number, E′E’E′ (complement) is rolling an odd number.

Laws of Probability

1. Addition Rule:
   • For mutually exclusive events: P(A∪B)=P(A)+P(B) 
   • For non-mutually exclusive events: P(A∪B)=P(A)+P(B)−P(A∩B) 
2. Multiplication Rule:
   • For independent events: P(A∩B)=P(A)×P(B) 
   • For dependent events: P(A∩B)=P(A)×P(B∣A) where P(B∣A) is the conditional probability of B given A.