Binomial properties

The binomial distribution counts successes in $n$ independent trials. Here we'll explore its key properties.

The shape of the binomial

The shape of a binomial distribution depends on $p$:

Try adjusting $n$ and $p$ in the explorer above. Notice how:

• When $p = 0.5$, the distribution is symmetric
• When $p < 0.5$, it's right-skewed (tail extends right)
• When $p > 0.5$, it's left-skewed (tail extends left)

Why E[X] = np

Where does the expected value formula come from?

We didn't need any complicated calculations. A binomial is just a sum of Bernoullis.

Adding binomials together

What happens when we add two independent binomial random variables?

This makes intuitive sense: $n$ trials plus $m$ more trials equals $n + m$ trials total.

Cumulative probabilities

Often we want $P(X \leq k)$ or $P(X \geq k)$, not just $P(X = k)$.

To calculate cumulative probabilities, we sum the individual probabilities:

$P(X \leq k) = \Large\sum_{i=0}^{k} P(X = i) = \Large\sum_{i=0}^{k} \binom{n}{i} p^i (1-p)^{n-i}$

When to use the binomial

The binomial distribution applies when you have:

1. Fixed number of trials $n$
2. Independence: each trial doesn't affect others
3. Same probability $p$ for each trial
4. Binary outcomes: success or failure

Test your understanding — can you identify which scenarios are truly Binomial?

Practice problems

What's next

What if we're sampling from a finite population without replacement? Then each draw changes the probabilities for future draws. That's the Hypergeometric distribution, coming up next.