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Fair Coin — Definition, Formula & Examples

A fair coin is a coin where each flip has exactly the same chance of landing heads as it does tails — a 50/50 split every time.

A fair coin is an idealized random device with two equally likely outcomes, heads (H) and tails (T), each occurring with probability 12\frac{1}{2} on every independent trial.

Key Formula

P(H)=P(T)=12P(H) = P(T) = \frac{1}{2}
Where:
  • P(H)P(H) = Probability of heads on a single flip
  • P(T)P(T) = Probability of tails on a single flip

How It Works

When a probability problem says "fair coin," it tells you to assign P(H)=P(T)=0.5P(H) = P(T) = 0.5. Each flip is independent, meaning past results do not change future probabilities. You can combine this per-flip probability with counting rules or the binomial formula to find the probability of any sequence or combination of outcomes over multiple flips.

Worked Example

Problem: A fair coin is flipped 3 times. What is the probability of getting exactly 2 heads?
Count total outcomes: Each flip has 2 outcomes, and the flips are independent.
Total outcomes=23=8\text{Total outcomes} = 2^3 = 8
Count favorable outcomes: List the sequences with exactly 2 heads: HHT, HTH, THH. Use the combination formula to confirm.
C(3,2)=3!2!⋅1!=3C(3,2) = \frac{3!}{2!\cdot1!} = 3
Compute the probability: Divide favorable outcomes by total outcomes, or apply the binomial formula with p=0.5p = 0.5.
P(exactly 2 heads)=C(3,2)(12)2(12)1=3⋅18=38P(\text{exactly 2 heads}) = C(3,2)\left(\frac{1}{2}\right)^2\left(\frac{1}{2}\right)^1 = 3 \cdot \frac{1}{8} = \frac{3}{8}
Answer: The probability of getting exactly 2 heads in 3 fair coin flips is 38=0.375\frac{3}{8} = 0.375.

Why It Matters

Nearly every introductory probability course uses fair coin flips to build intuition about independence, sample spaces, and the binomial distribution. Understanding the fair coin model also sets the stage for hypothesis testing in statistics, where you compare observed data against what a fair coin (or any null model) would predict.

Common Mistakes

Mistake: Believing that after several heads in a row, tails becomes more likely (the gambler's fallacy).
Correction: Each flip of a fair coin is independent. Previous outcomes have no effect on the next flip — the probability stays 12\frac{1}{2} every time.

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