Introduction to Probability

You already use probability every day without thinking about it. "There's no way it rains today." "I'll probably get a good seat if I show up early." "I've got maybe a 50/50 shot at this." What math does is take those fuzzy guesses and turn them into actual numbers.

The Scale Goes From 0 to 1

Here's the most important thing to know upfront: every probability is a number between 0 and 1.

A probability of 0 means something is impossible — it absolutely cannot happen. A probability of 1 means it's certain — it will definitely happen. Everything else lands somewhere in between.

So 0.5 means a 50-50 shot. Something with a probability of 0.9 is very likely. Something at 0.1? Don't count on it.

You'll see probabilities written three different ways — as decimals (0.5), fractions (\(\frac{1}{2}\)), or percentages (50%). They're all saying the exact same thing, just in different clothing. Use whichever form your problem calls for.

The Formula

When all possible outcomes are equally likely, finding a probability comes down to counting:

$$P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$$

One thing worth clarifying: "favorable" doesn't mean good. It just means the outcome you're interested in. If you're calculating the probability of rolling a 2, then rolling a 2 is your "favorable" outcome — even if 2 is a terrible roll in your game.

Starting Simple: Coins and Dice

Let's try the formula with something familiar. A fair coin has two sides — heads and tails — and both are equally likely.

What's the probability of flipping heads?

$$P(\text{heads}) = \frac{1}{2} = 0.5 = 50%$$

One favorable outcome (heads), two total outcomes (heads or tails). Nothing tricky there.

Now let's use a standard six-sided die. Each face is equally likely to land up.

Probability of rolling a 4:

$$P(4) = \frac{1}{6}$$

Only one face shows a 4, and there are six faces total.

Probability of rolling an even number:

The even numbers on a die are 2, 4, and 6 — three of them.

$$P(\text{even}) = \frac{3}{6} = \frac{1}{2}$$

Probability of rolling higher than 4:

Numbers higher than 4 are just 5 and 6.

$$P(\text{higher than 4}) = \frac{2}{6} = \frac{1}{3}$$

Notice that we always simplify at the end if we can.

Marbles in a Bag

Here's a classic setup you'll see a lot. A bag has 3 red marbles, 5 blue marbles, and 2 green ones. You reach in without looking and pull one out.

First, how many marbles are there total? \(3 + 5 + 2 = 10\). That's your denominator for everything.

Probability of drawing blue:

$$P(\text{blue}) = \frac{5}{10} = \frac{1}{2}$$

Probability of drawing red:

$$P(\text{red}) = \frac{3}{10}$$

Simple enough. Now here's where it gets a little more interesting.

The Complement: What Doesn't Happen

The complement of an event is just the flip side — all the outcomes where that event doesn't happen. And there's a really handy shortcut:

$$P(\text{not A}) = 1 - P(A)$$

This works because every outcome either is the event or isn't. Those two probabilities have to add up to 1.

Weather forecasters use this all the time. If there's a 30% chance of rain, there's automatically a 70% chance it stays dry. You don't need to calculate the dry probability separately — just subtract from 1.

Back to the marbles: what's the probability of not drawing green?

$$P(\text{not green}) = 1 - \frac{2}{10} = \frac{8}{10} = \frac{4}{5}$$

The complement trick is especially useful when it's easier to count what you don't want than what you do.

One Thing to Watch Out For

If your probability calculation ever gives you a number greater than 1 — say \(\frac{7}{4}\) — something went wrong. Probability can never exceed 1. Same goes for negatives. If you see either of those, it's worth backtracking to find the error before moving on.

Try These

1. A bag has 4 yellow and 6 purple chips. What is the probability of drawing yellow? Show answer\(\frac{2}{5}\) — 4 yellow out of 10 total: \(\frac{4}{10} = \frac{2}{5}\)

2. You roll a standard die. What is the probability of rolling a number less than 3? Show answer\(\frac{1}{3}\) — numbers less than 3 are 1 and 2: \(\frac{2}{6} = \frac{1}{3}\)

3. A jar has 2 red, 3 blue, and 5 white candies. What is the probability of not drawing white? Show answer\(\frac{1}{2}\) — 5 white out of 10 total, so \(P(\text{not white}) = 1 - \frac{5}{10} = \frac{1}{2}\)

4. A spinner has 8 equal sections numbered 1–8. What is the probability of landing on a multiple of 3? Show answer\(\frac{1}{4}\) — multiples of 3 from 1–8 are 3 and 6: \(\frac{2}{8} = \frac{1}{4}\)

5. If \(P(\text{event}) = 0.65\), what is the probability the event does not occur? Show answer0.35 — \(1 - 0.65 = 0.35\)