PROBABILITY

Probability is a branch of mathematics that deals with the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event.

Basic Terms in Probability

1. Experiment: A process that leads to an outcome.
2. Sample Space (S): The set of all possible outcomes of an experiment.
3. Event (E): A subset of the sample space.
4. Favorable Outcomes: The outcomes that satisfy a given event.
5. Probability Formula: Number of favorable outcomes / total number of outcomes in the sample space

Types of Probability

1. Theoretical Probability: Based on reasoning without actual experiments.
   • Example: The probability of getting heads when tossing a fair coin is 1/2.
2. Experimental Probability: Based on actual experiments and observations.
   • Example: If a die is rolled 100 times and the number 4 appears 20 times, the probability is 20/100 = 1/5.
3. Subjective Probability: Based on personal judgment and experience.
   • Example: The probability of a team winning a match based on expert opinion.

Rules of Probability

1. Addition Rule: For two mutually exclusive events A and B: P (A U B) = P(A) + P(B)
2. Multiplication Rule: For two independent events A and B: P ( A
3. Complementary Rule: The probability of an event not occurring:

Types of Events

1. Mutually Exclusive Events: Events that cannot occur at the same time.
2. Independent Events: The occurrence of one event does not affect the occurrence of another.
3. Dependent Events: The occurrence of one event affects the probability of another.
4. Complementary Events: The set of outcomes not in the event.

Applications of Probability

• Gambling & Games: Used in card games, dice games, and lotteries.
• Weather Forecasting: Predicting chances of rain, storms, etc.
• Finance & Insurance: Risk assessment in investments and policies.
• Medical Studies: Determining the likelihood of disease occurrence.
• Quality Control: Ensuring defect-free production in industries.

Example Problems

1. Coin Tossing: What is the probability of getting at least one head in two coin tosses?
   • Sample Space: HH, HT, TH, TT
   • Favorable Outcomes: HH, HT, TH (3 outcomes)
   • Probability:
2. Rolling a Die: What is the probability of rolling an even number?
   • Sample Space: {1, 2, 3, 4, 5, 6}
   • Favorable Outcomes: {2, 4, 6} (3 outcomes)
   • Probability: 3/6 = 1/2