01
Basic Concepts of Probability
Probability expresses the likelihood of an event occurring as a number between 0 and 1. Closer to 0 means less likely, closer to 1 means more likely. Basic probability is calculated as (favorable outcomes) ÷ (total outcomes). For example, the probability of rolling an even number on a die is 3/6 = 0.5 or 50%.
02
Independent and Dependent Events
Independent events occur when one event does not affect another. The probability of both independent events occurring is found by multiplying their probabilities. For example, the probability of flipping heads twice is 0.5 × 0.5 = 0.25. Dependent events occur when one event affects the probability of another, calculated using conditional probability.
03
Probability of Union Events
The probability of at least one of two events occurring is called union probability. If events cannot occur simultaneously (mutually exclusive), simply add their probabilities. Otherwise, use P(A or B) = P(A) + P(B) - P(A and B). Removing overlap is key.
04
Understanding Conditional Probability
Conditional probability is the probability of one event occurring given that another has occurred. Denoted P(B|A), meaning "probability of B given A". Used extensively in medical test accuracy, criminal investigations, spam filtering, and more. Bayes' theorem is a prominent application of conditional probability.
05
Real-World Applications
Probability is used in games, insurance, investment, weather forecasting, and more. Calculate odds of receiving specific poker hands, or insurance companies predict accident probabilities to set premiums. In stock investing, probability concepts are essential for evaluating expected values and risks.
06
Probability Calculation Tips
Break complex probability problems into steps. Accurately determine total outcomes and decide if order matters to use permutations or combinations appropriately. Using complement probability (1 - P) often simplifies calculations. For example, "probability of at least one success" equals 1 - "probability of all failures".
