Lesson · Probability

Introduction to Probability

Probability measures how likely an event is to occur. This lesson builds it up from one idea — counting outcomes — through multi‑step experiments and tree diagrams, to counting outcomes for experiments too big to draw.

1. What is probability?

To talk about the probability of an event, we first have to say exactly what situation the event happens in. If I ask “What is the probability of getting 6?” you have no way to answer — am I rolling a die? Drawing numbers from a hat? Counting goals in a soccer game? But if I am specific about the situation — the experiment — the question makes sense: “What is the probability of rolling a 6 on this 6‑sided die?” Now you can confidently answer 16.

Sometimes the experiment is specified but still can't be answered exactly. “What is the probability the US men's hockey team beats Finland at the next winter olympics?” is a real experiment, but so many variables feed into it that we could only ever approximate. For the work in this lesson we will always be able to compute the exact answer.

So how do we compute the probability of an event? It is the number of ways the experiment can produce that event, divided by the total number of possible outcomes.

Roll a standard 6‑sided die and there are 6 possible outcomes — 1, 2, 3, 4, 5, or 6 — each equally likely. So:

• Probability of rolling a 5? Just one outcome gives it, so 16.
• Probability of rolling an even number? Three outcomes work (2, 4, 6), so 36 = 12.
• Probability of rolling a 7? No outcome gives it, so 06 = 0.
• Probability of rolling a positive number? Every outcome gives it, so 66 = 1.

Look closely and you'll see a probability is always a number between 0 and 1. If the event can't happen — “draw 5 aces from this deck” — the top of the fraction is 0, so the whole thing is 0. If every outcome matches the event, the top equals the bottom, so it is xx = 1. Everything in between can happen but isn't guaranteed: the 2 of clubs comes up with probability 152, but you could just as easily draw the king of hearts. The closer to 1, the more likely; the closer to 0, the less.

2. Multi‑step experiments and tree diagrams

Experiments can have several steps — flip a coin, then roll a 6‑sided die. Now we can ask things like “What's the probability I flip heads and roll an odd number?” We answer it the same way: count the outcomes that match, divide by the total. To see every outcome at once, draw the experiment as a tree diagram.

Because each leaf is a single outcome, the questions become easy to answer:

• Heads and an odd number? Three leaves match — (H, 1), (H, 3), (H, 5) — so 312 = 14.
• Tails and a number under 3? Two leaves match — (T, 1), (T, 2) — so 212 = 16.

A bigger example

Roll a 4‑sided die, then a 3‑sided die, then a 2‑sided die. What's the probability of rolling all odd numbers? The tree is large, but it still just counts:

Counting the leaves reached through only odd branches gives 2 × 2 × 1 = 4 favorable outcomes, so:

3. Counting outcomes without drawing the tree

Drawing a tree isn't always practical — you'd never fit the tree for rolling a 10‑sided die 100 times on one page. Luckily we can count outcomes directly. Look again at the last example: we split into 4 nodes, each of those into 3, and each of those into 2. That's 4 buckets, each holding 3 buckets, each holding 2 things — which is just multiplication: 4 × 3 × 2 = 24.

The multiplication rule. In a multi‑step experiment where each step has a fixed number of equally‑likely outcomes, the total number of outcomes is the product of the outcomes at each step. Five steps with 3, 4, 2, 3, and 10 outcomes give 3 × 4 × 2 × 3 × 10 = 720.

Counting an event's outcomes

The same rule counts the outcomes for an event. Say we roll a 5‑sided die four times and want the probability of rolling all odd numbers. We only need to know how many odd outcomes one roll has — then multiply. Drawing just the first two layers shows why:

Each roll contributes 3 odd outcomes, so across four rolls there are 3⁴ = 81 favorable outcomes. Each roll has 5 outcomes total, so there are 5⁴ = 625 outcomes in all. The probability of rolling all odd numbers is:

Ready to practice?

These are exactly the questions the probability quiz drills — single rolls, a few rolls with a tree, and big dice too wide to draw. Or take the printable version with you.