Coin Tossing — Definition, Formula & Examples

Coin tossing is a probability experiment where you flip a fair coin and observe whether it lands on heads (H) or tails (T). Each flip has exactly two equally likely outcomes, making it the simplest model for studying randomness and chance.

A coin toss is a Bernoulli trial with sample space $S = \{H, T\}$ and probability $P(H) = P(T) = \tfrac{1}{2}$ for a fair coin. A sequence of $n$ independent coin tosses constitutes a binomial experiment with $2^n$ equally likely outcomes.

Key Formula

P(\text{exactly } k \text{ heads in } n \text{ flips}) = \binom{n}{k} \left(\frac{1}{2}\right)^n

Where:

$n$ = Total number of coin flips
$k$ = Number of heads you want
$\binom{n}{k}$ = Number of ways to choose which k flips are heads

How It Works

Each time you flip a fair coin, there is a

\tfrac{1}{2}

chance of heads and a

\tfrac{1}{2}

chance of tails. When you flip multiple times, the flips are independent — the result of one flip does not affect the next. To find the probability of a specific sequence (like HHT), you multiply the individual probabilities together. To count how many ways you can get exactly

k

heads in

n

flips, you use combinations:

\binom{n}{k}

Worked Example

Problem: You flip a fair coin 3 times. What is the probability of getting exactly 2 heads?

Step 1: Find the total number of equally likely outcomes for 3 flips.

2^3 = 8

Step 2: Count the number of ways to get exactly 2 heads out of 3 flips using combinations.

\binom{3}{2} = 3

Step 3: Divide favorable outcomes by total outcomes.

P(\text{2 heads}) = \frac{3}{8} = 0.375

Answer: The probability of getting exactly 2 heads in 3 flips is

\frac{3}{8}

, or 37.5%.

Visualization

Why It Matters

Coin tossing is the foundation for understanding binomial probability, which appears throughout algebra, statistics, and data science courses. Many real-world situations — like quality control (defective vs. non-defective) or medical testing (positive vs. negative) — follow the same two-outcome model.

Common Mistakes

Mistake: Believing that after several heads in a row, tails is "due" (the gambler's fallacy).

Correction: Each flip is independent. Past results have no effect on the next flip — the probability stays

\frac{1}{2}

every time.