Probability Basics

Probability is a branch of mathematics that deals with the likelihood of an event occurring. It is a way of quantifying uncertainty and measuring the likelihood of different outcomes. In the context of coin flips, probability is used to determine the likelihood of the coin landing on heads or tails.

This article is an introduction to the topic of probability. We will stick to plain language and use a coin as the only example. More advanced statistics will come in future articles.

The Basics

When we talk about probability, this is a mathematical concept that can be represented in a few different ways. The most common way to represent probability is as a fraction, decimal, or percentage. For example, if you flip a fair coin, the probability of it landing on heads is $\frac{1}{2}$, which is the same as 0.5, or also 50%. The three representations are equivalent and can be used interchangeably.

As we will see in The Unexpected Statistics of Coin Flips blog post, the probability of a coin landing is not exactly 50-50 every time.

Outcomes and events

Before we go further, two small pieces of vocabulary:

• An outcome is one of the possible things that can happen. When you flip a coin, the outcomes are heads, tails, and (very rarely) edge.
• An event is a question we ask about the outcomes, like "did it land on heads?" or "did it land on a face at all?".

Probability is the tool we use to put a number on how likely an event is.

The probability scale

Probabilities always live on a scale from 0 to 1:

• 0 means the event is impossible. A coin turning into a pigeon mid-air has probability 0.
• 1 means the event is certain. A flipped coin eventually coming back down, thanks to gravity, has probability 1.
• Anything in between is "somewhat likely". A fair coin landing on heads sits right in the middle, at 0.5.

In percentage terms, that same scale goes from 0% to 100%. Same idea, expressed in a different way.

The complement rule

Every event has a mirror image called its complement: the event of it not happening. And because something either happens or it doesn't, the two probabilities always add up to 1. In symbols:

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

For a fair coin:

• $P(\text{heads}) = 0.5$, so $P(\text{not heads}) = 1 - 0.5 = 0.5$.
• $P(\text{face}) \approx 1 - \tfrac{1}{6000} \approx 0.99983$, because the only other option is the rare "edge" landing. (More on that 1-in-6000 number in The Unexpected Statistics of Coin Flips.)

The complement rule is often the easiest way to compute a probability: instead of counting every way something can happen, count the one way it can't and subtract from 1.

Combining events: "and" vs "or"

Things get interesting when we ask about more than one flip at a time. Two rules cover most of what you'll ever need:

"AND" — both things happen

Coin flips don't influence each other (they are independent). In this case, you multiply their probabilities:

• $P(\text{OO}) = 0.5 \times 0.5 = 0.25$, or 25%.
• $P(\text{OOO}) = 0.5 \times 0.5 \times 0.5 = 0.125$, or 12.5%.

Each extra flip halves the probability. That's why long streaks feel surprising — not because the coin is "due" for the other side, but because each new flip piles another $\tfrac{1}{2}$ onto the product.

"OR" — at least one of several things happens

If two outcomes can't happen at the same time (they're mutually exclusive), you add their probabilities:

• $P(\text{O} \cup \text{X}) = 0.5 + 0.5 = 1$. Unsurprising — that's every non-edge outcome.
• $P(1 \cup 2) = \tfrac{1}{6} + \tfrac{1}{6} = \tfrac{2}{6}$ (rolling a 1 or a 2 on a six-sided die).

If the events can overlap, simple addition double-counts the overlap and you have to subtract it back out:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

That's the inclusion–exclusion principle, and the deeper version is a topic for a future article.

A quick note on independence

Two events are independent when the outcome of one doesn't change the probability of the other. Coin flips are the textbook example: the coin has no memory, so a run of ten heads in a row doesn't make tails any more likely on the eleventh flip.

This is the setup behind the famous Gambler's Fallacy, which I unpack in the statistics article. For now, the short version: each flip is a fresh start.

Conclusion

That's really it for the basics:

1. Probabilities live between 0 and 1.
2. Use the complement (1 − P) when it's easier to count what doesn't happen.
3. Multiply for "and", add for mutually exclusive "or".
4. Independent events don't care about each other's history.

With just those four ideas you can reason about almost any coin-flip question — and most dice, cards, and lottery-style problems too.

Put the theory to work: flip a coin online and track your own streak.