What is Probability?
Probability is a number that measures how likely an event is to happen. It's expressed as a value between 0 and 1, where 0 means the event is impossible and 1 means it's certain. The probability of an event is calculated by dividing the number of favourable outcomes by the total number of possible outcomes.
Probability Formula
The probability of an event E is given by:
P(E) = n(E) / n(S)
In plain language, probability equals the number of favourable outcomes divided by the total number of outcomes in the sample space.
Symbol | Meaning |
|---|---|
P(E) | Probability of event E |
n(E) | Number of favourable outcomes |
n(S) | Total number of outcomes (sample space) |
E | The event being measured |
S | The sample space β the set of all possible outcomes |
The formula assumes all outcomes in the sample space are equally likely. This is the case for fair coins, fair dice, well-shuffled card decks, and most introductory probability problems.
Formal Definition
Formally, probability is a function P that assigns a real number between 0 and 1 to every event in a sample space, satisfying three conditions known as Kolmogorov's axioms: P(E) β₯ 0 for any event E, P(S) = 1 (some outcome in the sample space must occur), and P(A βͺ B) = P(A) + P(B) for any two mutually exclusive events A and B. These three rules form the foundation of modern probability theory.
Probability Range and Scale
Probability values fall on a scale from 0 to 1. Four boundary cases describe what each end of that scale means:
P(E) = 0 β the event is impossible (rolling a 7 on a six-sided die)
P(E) = 1 β the event is certain (rolling a number less than 7 on a six-sided die)
P(E) = 0.5 β the event is equally likely to occur or not occur (a fair coin landing heads)
0 < P(E) < 1 β the event is uncertain, with the value indicating how likely it is
The complement rule connects an event to its opposite: P(E) + P(E') = 1. If the probability of rain tomorrow is 0.3, the probability of no rain is 0.7.
Worked Example
Find the probability of rolling a number greater than 4 on a standard six-sided die.
Sample space: S = {1, 2, 3, 4, 5, 6}, so n(S) = 6 Favourable outcomes: E = {5, 6}, so n(E) = 2
P(E) = n(E) / n(S) = 2/6 = 1/3
Answer: P(rolling a number greater than 4) = 1/3
Expressed as a decimal, this is approximately 0.333. Expressed as a percentage, 33.3%.
Types of Probability
There are three main types of probability used in mathematics. Each defines probability in a slightly different way depending on what information is available.
Theoretical Probability
Theoretical probability is calculated by reasoning about equally likely outcomes β without performing any experiment. The formula P(E) = n(E) / n(S) gives theoretical probability. Tossing a fair coin has a theoretical probability of 1/2 for heads.
Experimental Probability
Experimental probability (also called empirical probability) is calculated from the actual results of repeated trials. The formula is P(E) = (number of times E occurred) / (total number of trials). If a coin lands heads 47 times out of 100 tosses, the experimental probability of heads is 47/100 = 0.47.
Axiomatic Probability
Axiomatic probability is a formal framework defined by Kolmogorov's three axioms. It applies to all probability problems β both theoretical and experimental β and forms the basis of advanced probability theory.
Conditional probability and subjective probability are two further extensions, covered separately.
Probability Notation Reference
Notation | Meaning |
|---|---|
P(A) | Probability of event A |
P(A') or P(AΜ ) | Probability of A not occurring (complement of A) |
P(A βͺ B) | Probability of A or B (or both) β union |
P(A β© B) | Probability of A and B both occurring β intersection |
P(A | B) | Probability of A given that B has occurred β conditional |
n(E) | Number of outcomes in event E |
n(S) | Total number of outcomes in the sample space |
These symbols appear across probability problems at every level - from elementary school through to university statistics.
Related Terms in Probability
Term | Meaning | How It Relates |
|---|---|---|
Experiment | An action with an uncertain outcome (rolling a die, drawing a card) | The setup that generates a probability question |
Outcome | A possible result of one trial | A building block of the sample space |
Sample space (S) | The set of all possible outcomes | Forms the denominator in the probability formula |
Event (E) | A subset of the sample space β one or more outcomes | Forms the numerator in the probability formula |
Trial | One performance of the experiment | Repeated trials build experimental probability |
Equally likely outcomes | Outcomes with the same chance of occurring | Required for theoretical probability calculations |
Mutually exclusive events | Two events that cannot occur together | P(A β© B) = 0 |
Complementary events | An event and its "not happening" counterpart | P(A) + P(A') = 1 |
Common Confusions About Probability
Probability vs odds. Probability is favourable outcomes divided by total outcomes. Odds are favourable outcomes compared to unfavourable outcomes. If P(E) = 1/4, the odds are 1 to 3 β not 1 to 4. Probability is a single number on a 0-to-1 scale; odds are a ratio.
Probability vs chance vs possibility. In everyday language, these words are often interchangeable. In mathematics, only probability has a precise numerical definition on the 0-to-1 scale. "Chance" and "possibility" are informal terms β useful in conversation, not in calculation.
The "between 0 and 1" misconception. Some students assume probability cannot equal exactly 0 or exactly 1. Both bounds are valid. P(E) = 0 describes an impossible event; P(E) = 1 describes a certain event. The scale is inclusive at both ends β written formally as 0 β€ P(E) β€ 1.
When You'll See Probability
Probability appears in school curricula across most international systems:
CCSS (United States): Probability is introduced in Grade 7 (7.SP.5, 7.SP.6, 7.SP.7) and extended in high school statistics
NCERT (India): Class 9 Chapter 14 covers empirical probability; Class 10 Chapter 15 covers theoretical probability; Class 11 Chapter 16 introduces sets and axiomatic probability
UK National Curriculum: Probability scales appear in Key Stage 3; conditional probability appears in GCSE and A-level mathematics
Cambridge IGCSE: Probability is covered in the core and extended syllabuses
Beyond school, probability is the foundation of weather forecasting, insurance pricing, genetics, machine learning, quality control, and most fields that involve uncertainty.
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