2.4 Probability Concepts - 2.4 Probability Concepts
Key Concepts
- Probability Basics
- Conditional Probability
- Bayes' Theorem
- Random Variables and Probability Distributions
Probability Basics
Probability is the measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The basic formula for probability is:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
Example: The probability of rolling a 3 on a six-sided die is \( \frac{1}{6} \), as there is one favorable outcome (rolling a 3) out of six possible outcomes.
Conditional Probability
Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as \( P(A|B) \), which reads as "the probability of A given B." The formula for conditional probability is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Example: If the probability of rain today is 40% and the probability of rain and traffic is 20%, the conditional probability of traffic given rain is \( \frac{0.20}{0.40} = 0.50 \) or 50%.
Bayes' Theorem
Bayes' Theorem is a formula that provides a way to update the probability of a hypothesis based on new evidence. It is particularly useful in situations where we need to revise our beliefs in light of new data. The formula for Bayes' Theorem is:
\[ P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)} \]
Example: Suppose a medical test for a disease is 99% accurate. If the disease prevalence is 1%, and a person tests positive, the probability that the person actually has the disease is calculated using Bayes' Theorem.
Random Variables and Probability Distributions
A random variable is a variable whose value is determined by the outcome of a random event. Probability distributions describe the likelihood of different outcomes for a random variable. There are two main types: discrete and continuous.
Example: A discrete random variable could be the number of heads in three coin flips, with a probability distribution showing the likelihood of 0, 1, 2, or 3 heads. A continuous random variable could be the height of individuals in a population, with a probability distribution showing the likelihood of heights within a certain range.