Understanding the Basics of Probability
1. Basic Vocabulary
Probability is a branch of mathematics that deals with the likelihood of different outcomes. A random experiment is an experiment whose outcome cannot be predicted with certainty. Examples include tossing a coin, rolling a die, or drawing a card.
2. Sample Space (Ω)
The sample space is the set of all possible outcomes of a random experiment. For example, when tossing a coin, the sample space is Ω = {Heads, Tails}. For a die, it is Ω = {1, 2, 3, 4, 5, 6}. An event is a subset of the sample space, such as getting an even number when rolling a die, which would be {2, 4, 6}.
3. Probability of an Event
The probability of an event A, denoted P(A), is calculated as the number of favorable outcomes divided by the total number of possible outcomes. For example, the probability of rolling a 3 on a die is 1/6.
4. Important Properties
Probability has several important properties:
- 0 ≤ P(A) ≤ 1
- P(Ω) = 1
- P(∅) = 0
5. Addition of Probabilities
For mutually exclusive events A and B, the probability of A or B occurring is the sum of their probabilities: P(A ∪ B) = P(A) + P(B). For example, if A is getting a 1 and B is getting a 6 on a die, then P(A ∪ B) = 1/6 + 1/6 = 1/3.
6. Probability & Multiplication
For independent events A and B, the probability of both A and B occurring is the product of their probabilities: P(A ∩ B) = P(A) × P(B). For example, the probability of rolling two 6s with two dice is 1/6 × 1/6 = 1/36.
7. Conditional Probability
Conditional probability is used when the outcome of one event affects the outcome of another. The probability of A given B is denoted P(A|B) and is calculated as P(A ∩ B) / P(B).
8. Binomial Distribution
The binomial distribution is used to model the number of successes in a fixed number of independent trials, each with the same probability of success. It is characterized by the parameters n (number of trials) and p (probability of success).
Quick Summary
- Simple Cases: Favorable / Possible
- Complement: 1 - P(A)
- Independent: P(A) × P(B)
- Conditional: P(A ∩ B) / P(A)