Introduction to Probability

Introduction to Probability

• Probability is a measure of the likelihood of an event occurring.
• It's expressed as a number between 0 and 1, inclusive.
• A probability of 0 indicates an impossible event.
• A probability of 1 indicates a certain event.

Defining Probability

• Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
• This is often expressed as P(event) = favorable outcomes / total outcomes.
• The probability of an event that is expected to occur is higher than an event that is expected not to occur.

Types of Probability

• Theoretical Probability: This is the likelihood of an event occurring based on the possible outcomes. A fair coin has a 50% theoretical probability of landing on heads.
• Experimental Probability: This is the likelihood based on observed trials. Flipping a coin 100 times and getting 52 heads gives an experimental probability of 52/100 = 52%. This may not match the theoretical probability if the experiment doesn't accurately reflect the underlying process (e.g. a biased coin).

Complementary Events

• Complementary events are two outcomes that are mutually exclusive (they can't happen at the same time).
• The sum of the probabilities of complementary events is always 1. If the probability of rain is 20%, the probability of no rain is 80%.

Independent Events

• Independent events are events where the outcome of one event does not affect the outcome of another.
• The probability of multiple independent events occurring together is the product of their individual probabilities. Flipping a coin twice and getting heads both times is (1/2) * (1/2) = 1/4 = 25%.

Dependent Events

• Dependent events are events where the outcome of one event affects the outcome of another.
• The probability of multiple dependent events occurring together is more complex to calculate than independent events. It requires considering how the first event changes the possible outcomes for the second event.

Mutually Exclusive Events

• Mutually exclusive events cannot occur at the same time.
• The probability of either one of two mutually exclusive outcomes occurring is the sum of their individual probabilities. Drawing a red or blue card from a deck of cards, assuming there are equal numbers of each, would have a probability of red + probability of blue.

Conditional Probability

• Conditional Probability is the probability of one event occurring, given that another event has already occurred.
• The probability of event A given event B has occurred is denoted as P(A|B).
• The formula for conditional probability is P(A|B) = P(A and B) / P(B).

Sample Space

• The sample space is the set of all possible outcomes of an experiment or event.
• The sample space is important in probability theory as it lays the groundwork for calculating probabilities.

Basic Probability Rules

• The probability of an event must be between 0 and 1.
• The sum of the probabilities of all possible outcomes in a sample space is equal to 1.