PROBABILITY / 4 MINUTE READ

Coin flip probability, minus the mystery.

A fair coin has two possible outcomes, but repeated flips create patterns that often feel anything but fair. Here is how to understand the odds without mistaking a streak for a broken coin.

The probability of one coin flip

For an ideal fair coin, Heads and Tails each have a probability of one half, written as 1/2, 0.5, or 50%. The two probabilities add to 100% because one of those outcomes must occur. A virtual coin can provide the same equal odds by assigning half of its possible random values to Heads and half to Tails.

Equal probability does not mean results must alternate. Each flip is independent: the next outcome is unaffected by earlier flips. After five Heads in a row, the probability of Heads on the sixth fair flip is still 50%.

Why streaks are normal

People are excellent at finding patterns, including patterns that randomness produces naturally. In ten flips, a run of three identical results is not surprising. There are 1,024 possible ordered sequences of ten Heads-or-Tails outcomes, and many contain visible streaks.

The probability of getting Heads five times in a row is (1/2)⁵, or 1 in 32. That is uncommon in a single five-flip attempt, but it becomes much less surprising when many people are flipping many coins.

Expected results over time

As the number of flips grows, the proportion of Heads usually moves closer to 50%. This is the law of large numbers. It does not promise exactly half Heads in every sample. A set of 100 flips might contain 46 Heads; a much larger set will usually show a smaller percentage difference from the expected value.

Useful formulas

• A specific sequence of n flips: probability = (1/2)ⁿ.
• All possible sequences: 2ⁿ.
• At least one Head in n flips: 1 − (1/2)ⁿ.

These formulas assume independent flips and a fair coin. Physical coins can have tiny biases, while a properly implemented online tool can divide its random output evenly.

When a coin flip is useful

A coin toss is excellent for low-stakes binary choices, choosing who starts, breaking a tie, or demonstrating probability. It should not decide medical, legal, financial, safety, or other consequential questions. Randomness removes preference; it does not add judgment.