What is Probability?
Probability measures how likely an event is to happen, expressed as a number between 0 and 1. It's the mathematical language for chance, used everywhere from games and weather forecasts to genetics and insurance.
Probability is the ratio of favorable outcomes to total possible outcomes: P(A) = n(A)/n(S). It ranges from 0 (impossible) to 1 (certain).
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Step-by-step worked examples
A fair six-sided die is rolled once. What is the probability of rolling a 4?
Sample space: {1,2,3,4,5,6}, so n(S) = 6
Favorable outcome: {4}, so n(A) = 1
P(A) = n(A)/n(S) = 1/6 ≈ 0.167A standard deck of 52 cards is shuffled. What is the probability of drawing a heart?
Total cards: n(S) = 52 Hearts in the deck: n(A) = 13 P(A) = 13/52 = 1/4 = 0.25
Two fair coins are flipped. What is the probability both land on heads?
Sample space: {HH, HT, TH, TT}, so n(S) = 4
Favorable outcome: {HH}, so n(A) = 1
P(A) = 1/4 = 0.25Flashcards
Quick quiz
Q1.A bag has 4 red and 6 blue marbles. What is the probability of picking a red marble?
Q2.What is the probability of an impossible event?
Q3.Rolling a standard die, what is P(number > 4)?
Q4.If P(A) = 0.3, what is the probability that A does NOT happen?
The full card deck, worked steps and AI-tutor support for “What is Probability?” are in Notek — study by hand before your exam.
Common mistakes
Thinking probability can be greater than 1. — Correct: Probability is always between 0 and 1 — a value like 1.5 signals a calculation error.
Forgetting to count all possible outcomes in n(S). — Correct: List the full sample space carefully before counting favorable outcomes.
Confusing 'or' (union) with 'and' (intersection) events. — Correct: 'Or' usually means adding probabilities; 'and' usually means multiplying (for independent events).
Assuming past results affect independent future events (gambler's fallacy). — Correct: For independent events like coin flips, each trial has the same probability regardless of history.
FAQ
What is the formula for probability?
P(A) = n(A)/n(S) — the number of favorable outcomes divided by the total number of possible outcomes.
What is probability in simple terms?
It's a measure of how likely something is to happen, from 0 (never) to 1 (always).
How to calculate probability of an event?
Count the favorable outcomes, count the total outcomes, then divide favorable by total.
What are some real-life examples of probability?
Weather forecasts, card games, dice rolls, genetics, insurance risk, and lottery odds all use probability.