Probability
Basic Concepts
Probability measures how likely an event is to occur. It ranges from 0 (impossible) to 1 (certain).
Probability of an event A is denoted as P(A).
If all outcomes are equally likely:
P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
Example 1: What is the probability of rolling a 4 on a fair six-sided die?
Probability Rules
- 0 \leq P(A) \leq 1 for any event A.
- P(\emptyset) = 0 (impossible event)
- P(S) = 1 where S is the sample space (certain event).
- P(A^c) = 1 - P(A) where A^c is the complement of A.
- P(A \cup B) = P(A) + P(B) - P(A \cap B) (Addition rule)
Example 2: Given P(A) = 0.3, P(B) = 0.5, and P(A \cap B) = 0.2, find P(A \cup B).
Conditional Probability
Conditional probability is the probability of event A given event B has occurred, denoted:
P(A \mid B) = \frac{P(A \cap B)}{P(B)}, where P(B) > 0.
Example 3: If P(A \cap B) = 0.15 and P(B) = 0.3, find P(A \mid B).
Discrete Probability Distributions
A discrete random variable takes countable values. Probability distribution lists probabilities for each value.
Example: Tossing a fair coin twice, possible outcomes and probabilities:
- HH: P = \frac{1}{4}
- HT: P = \frac{1}{4}
- TH: P = \frac{1}{4}
- TT: P = \frac{1}{4}
Example 4: What is the probability of getting exactly one head when tossing two coins?
Practice Problems
Problem 1: A box contains 3 red, 5 blue, and 2 green balls. What is the probability of picking a red ball?
Problem 2: Two dice are rolled. What is the probability the sum is 7?
Problem 3: If P(A) = 0.4 and P(B) = 0.5, and they are mutually exclusive, find P(A \cup B).