What Is Probability?

Probability is the mathematics of uncertainty. It gives us a way to reason about events we cannot predict with certainty — which is most of life. Expressed as a number between 0 and 1 (or 0% and 100%), probability tells us how likely something is to occur.

  • 0 (0%) = impossible
  • 0.5 (50%) = equally likely to happen or not
  • 1 (100%) = certain

The probability of a fair coin landing heads is 0.5. The probability of rolling a 3 on a standard die is 1/6, or about 0.167 (16.7%).

The Two Types of Probability

Theoretical Probability

This is calculated from known, equally likely outcomes. If a bag contains 4 red marbles and 6 blue marbles, the probability of drawing a red marble is 4/10 = 0.4 (40%). No experiment needed — we calculate it from the known composition.

Empirical (Experimental) Probability

This is based on observed results. If you flip a coin 1,000 times and get 487 heads, your empirical probability is 487/1000 = 48.7%. With enough data, empirical probability converges toward theoretical probability — this is called the Law of Large Numbers.

Independent vs. Dependent Events

Two events are independent if the outcome of one has no effect on the other. A coin flip is always independent — getting heads ten times in a row does not make tails "more likely" on the next flip. This misunderstanding is called the Gambler's Fallacy.

Two events are dependent if one affects the other. Drawing a card from a deck and not replacing it changes the probabilities of subsequent draws.

Conditional Probability: The Key to Nuanced Thinking

Conditional probability asks: given that something has already happened, how likely is something else? Written as P(A|B) — "the probability of A given B."

This concept is crucial for medical testing. Suppose a test for a disease is 99% accurate. A person tests positive. What is the probability they actually have the disease? The answer depends heavily on how common the disease is in the population (its base rate). If the disease affects 1 in 10,000 people, a positive test result may still be more likely to be a false positive than a true positive. This is why doctors consider context, not just test results.

Expected Value: The Rational Decision-Maker's Tool

Expected value (EV) is the average outcome you'd expect if you repeated a decision many times. It's calculated by multiplying each possible outcome by its probability and summing them.

Example: A lottery ticket costs Rp 10,000. The prize is Rp 500,000 with a 1% chance of winning. EV = (0.01 × 500,000) + (0.99 × 0) = Rp 5,000. Since the ticket costs Rp 10,000 but the expected return is only Rp 5,000, buying it is a poor financial decision on average.

Practical Applications in Daily Life

  1. Insurance: Understanding that low-probability, high-impact events justify paying a premium even when you "expect" to lose money on the policy.
  2. Medical decisions: Evaluating screening tests and treatment options based on actual probabilities, not just "it could help."
  3. Business planning: Assigning probabilities to different market scenarios to make more robust plans.
  4. Everyday risk: Deciding whether to carry an umbrella based on a weather forecast that says "40% chance of rain."

Getting Comfortable with Uncertainty

The goal of learning probability isn't to eliminate uncertainty — it's to navigate it more wisely. The world is irreducibly unpredictable. Probability gives you a framework to make rational, informed choices even when outcomes cannot be guaranteed. In a world full of noise and chance, that framework is enormously valuable.