Binomial distribution: complete lesson (Bernoulli, formula, expectation)

Bernoulli trial

A Bernoulli trial is a random experiment with 2 outcomes:

• “Success” with probability $p$.
• “Failure” with probability $q = 1 - p$.

Examples: coin flip, drawing a card (red or black), testing if an item works.

Bernoulli scheme

$n$ Bernoulli trials:

• Identical (same $p$).
• Independent.

Let $X$ be the number of successes. Then $X$ follows a binomial distribution with parameters $n$ and $p$: $X \sim \mathcal{B}(n, p)$.

Binomial formula

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

for $k = 0, 1, 2, \ldots, n$.

Interpretation:

• $\binom{n}{k}$: number of ways to get $k$ successes among $n$ slots.
• $p^k$: probability of $k$ successes.
• $(1-p)^{n-k}$: probability of $n-k$ failures.

Expectation and variance

$E(X) = n \cdot p$

$V(X) = n \cdot p \cdot (1 - p)$

$\sigma(X) = \sqrt{n p (1 - p)}$

Meaning: over $n$ trials with success probability $p$, expect $np$ successes.

Typical example

Flip a fair coin 10 times. Probability of exactly 3 heads?

$X \sim \mathcal{B}(10, 0.5)$, $k = 3$.

$P(X = 3) = \binom{10}{3} \times 0.5^3 \times 0.5^7 = 120 \times 0.5^{10} \approx 0.117$

Cumulative: $P(X \leq k)$

For an inequality, sum up:

$P(X \leq k) = P(X = 0) + P(X = 1) + \ldots + P(X = k)$

For $P(X \geq k) = 1 - P(X \leq k - 1)$ (complement).

Graph

Bar chart: heights = $P(X = k)$. Symmetric if $p = 0.5$, skewed otherwise.

Special cases

• $n = 1$: $X$ is a Bernoulli variable ($X \in \{0, 1\}$). $E(X) = p$, $V(X) = p(1-p)$.
• $p = 0$: $X = 0$ always.
• $p = 1$: $X = n$ always.

Application: normal approximation

For large $n$ with $p$ not too close to $0$ or $1$, the binomial approaches a normal distribution with mean $np$ and standard deviation $\sqrt{np(1-p)}$.

Common mistakes

1. Forgetting independence of trials.
2. Writing $E(X) = p$ instead of $np$.
3. Confusing $\binom{n}{k}$ with $n^k$ in the formula.
4. Using binomial when trials aren’t identical (e.g., drawing without replacement = hypergeometric).