Probability Basics

Probability is the math of "maybe." It's how we put a number on how likely something is to happen. Weather forecasts ("30% chance of rain"), medical risks ("this treatment works for 70% of patients"), and game design all rely on probability. Understanding the basics is one of the most useful life skills math gives you.

**Probability** is a number between 0 and 1 that tells you how likely an event is:\n\n- **0** = impossible (will never happen)\n- **1** = certain (will always happen)\n- **0.5** = even chance (50% likely)\n\nProbabilities can be written as decimals (0.5), fractions (1/2), or percents (50%) — all the same thing.\n\nFormula for simple probability:\n\n**P(event) = (number of favorable outcomes) / (total number of equally likely outcomes)**\n\nCoin flip: P(heads) = 1/2 (one favorable: heads; total: 2 sides).\n\nRolling a die: P(rolling a 3) = 1/6.\n\nRolling an even number: P(even) = 3/6 = 1/2 (favorable: 2, 4, 6).

**Theoretical probability** = what SHOULD happen based on the setup. P(heads on a fair coin) = 1/2 in theory.\n\n**Experimental probability** = what DID happen when you tried it. If you flip a coin 10 times and get 6 heads, experimental P(heads) = 6/10 = 0.6.\n\nThe more times you repeat an experiment, the closer experimental probability gets to theoretical. Flip a coin 10,000 times, you'll get very close to 5,000 heads. This is called the **Law of Large Numbers**.

The **complement** of an event is everything that is NOT that event.\n\nIf P(rain today) = 0.3, then P(NOT rain today) = 0.7.\n\n**P(A) + P(NOT A) = 1**\n\nThis is useful when "not A" is easier to count. If you want to know P(at least one head in 3 coin flips), easier to compute P(no heads = all tails) and subtract from 1.\n\nP(all tails) = (1/2)³ = 1/8. So P(at least one head) = 1 − 1/8 = 7/8.

**Independent events**: the outcome of one doesn't affect the other. Two coin flips are independent. Two die rolls are independent.\n\nFor independent events: P(A AND B) = P(A) × P(B).\n\nP(two heads in a row) = 1/2 × 1/2 = 1/4.\n\n**Dependent events**: the outcome of one DOES affect the other. Drawing two cards from a deck without replacing the first one: the second draw's probabilities shift because the deck has changed.

When events can happen in multiple ways, count the possibilities. Tools:\n\n- **Tree diagrams**: branches for each outcome.\n- **Tables**: rows × columns.\n- **Sample space lists**: write out every possible combo.\n\nExample: rolling two dice. There are 6 × 6 = 36 equally likely outcomes. Probability of rolling a sum of 7? There are 6 ways to get 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). So P(sum = 7) = 6/36 = 1/6. Probability of rolling a sum of 2? Only one way (1+1). So P(sum = 2) = 1/36.

Probability is everywhere — from deciding whether to pack an umbrella to understanding medical test results. Once you start thinking in probabilities, you'll be less fooled by claims like "99% effective!" (at what? over what sample?) and better at making real decisions under uncertainty. It's one of the most practically useful parts of math in adult life.