Probability Basics

What Are the Chances?

Probability is a way to measure the likelihood of something happening. It's a number between 0 and 1, where 0 means the event is impossible and 1 means it's absolutely certain. You'll often see probabilities written as fractions, decimals, or percentages.

For any event, which we can call $E$ , we can find its probability using a simple formula, as long as all outcomes are equally likely:

P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}

Think about rolling a standard six-sided die. There are six possible outcomes: 1, 2, 3, 4, 5, or 6. What's the probability of rolling a 3? There's only one way for that to happen, out of six total possibilities. So, the probability is $1/6$.

What about the probability of rolling an even number? The favorable outcomes are 2, 4, and 6. That's three possibilities. The probability is $3/6$, which simplifies to $1/2$.

All the Possibilities

To calculate probabilities accurately, we need to be clear about all the things that could happen. This brings us to two key ideas: sample spaces and events.

Sample Space

noun

The set of all possible outcomes of an experiment. It is usually denoted by S.

Event

noun

A specific outcome or a set of outcomes from an experiment. An event is a subset of the sample space.

Let's use the example of drawing a single card from a standard 52-card deck.

The sample space is the entire set of 52 cards.

An event could be:

Drawing a King
Drawing a Heart
Drawing the Ace of Spades

The Rules of the Game

Probability isn't just guesswork; it's built on three solid rules, called axioms. These were formalized by the mathematician Andrey Kolmogorov. Everything else in probability theory can be derived from these three ideas.

Axiom 1: Non-negativity The probability of any event is always non-negative. It's either zero or positive. You can't have a negative chance of something happening. For any event A, $P(A) \geq 0$ .

Axiom 2: Certainty The probability of the entire sample space is 1. Something from the set of all possible outcomes must occur. $P(S) = 1$ .

Axiom 3: Additivity If two events A and B are mutually exclusive (meaning they can't both happen at the same time), the probability of either A or B occurring is the sum of their individual probabilities. $P(A \text{ or } B) = P(A) + P(B)$ .

From these axioms, we can figure out some other useful rules. For example, what's the probability of an event not happening? This is called the complement of an event, written as $A^c$ or $A'$ .

Since an event either happens or it doesn't, the probability of it happening plus the probability of it not happening must equal 1 (certainty).

P(A) + P(A^c) = 1

Rearranging this gives us the complement rule:

P(A^c) = 1 - P(A)

This rule is surprisingly handy. Sometimes it's much easier to calculate the probability of an event not happening than the probability of it happening.

For example, what's the probability of not rolling a 6 on a single die? We know the probability of rolling a 6 is $1/6$. So, the probability of not rolling a 6 is: $1 - 1/6 = 5/6$.

Let's check your understanding of these basic concepts.

Quiz Questions 1/5

If the probability of an event is 1, what does this signify?

Quiz Questions 2/5

When rolling a standard six-sided die, what is the probability of rolling a number greater than 4?

These fundamental concepts are the building blocks for understanding risk and making decisions in the face of uncertainty.

Probability for Everyday Decisions