Binomial Distribution: Coin Flips and Success Counts

What is the binomial distribution?

The binomial distribution answers: "If I repeat an experiment $n$ times, each with probability $p$ of success, what's the probability of getting exactly $k$ successes?"

The formula

$P(X = k) = \binom{n}{k} p^{k}(1-p)^{n-k}$

where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is "n choose k" — the number of ways to pick which $k$ of the $n$ trials are successes.

Step-by-step example: $n = 20$ coin flips, $p = 0.5$

What is $P(\text{exactly 10 heads})$?

Step 1 — Identify values: $n = 20$, $k = 10$, $p = 0.5$.

Step 2 — Compute $\binom{20}{10}$: $\binom{20}{10} = 184{,}756$.

Step 3 — Apply the formula:

$P(X = 10) = 184{,}756 \times (0.5)^{10} \times (0.5)^{10} = 184{,}756 \times (0.5)^{20}$

$= 184{,}756 / 1{,}048{,}576 \approx 0.176$

So there's about a 17.6% chance of getting exactly 10 heads in 20 flips.

Key properties

• Mean: $\mu = np = 20 \times 0.5 = 10$
• Standard deviation: $\sigma = \sqrt{np(1-p)} = \sqrt{20 \times 0.5 \times 0.5} = \sqrt{5} \approx 2.24$
• Shape: Symmetric when $p = 0.5$; skewed left when $p > 0.5$; skewed right when $p < 0.5$.

When to use the binomial

The binomial applies when: (1) fixed number of trials $n$, (2) each trial has exactly two outcomes (success/failure), (3) constant probability $p$, (4) trials are independent.

Examples: coin flips, defective items in a batch, free throw success rate, true/false quiz guessing.

Try it in the visualization

Adjust $n$ and $p$. The bar chart shows $P(X = k)$ for each $k$. The mean line shifts with $np$. Toggle the normal approximation to see how the binomial becomes bell-shaped for large $n$.