How to find binomial probability

Answer

Binomial probability is found using the binomial probability mass function, which gives the chance of getting exactly $x$ successes in $n$ independent trials with success probability $p$. You can also add several of these probabilities to get cumulative probabilities like “at most” or “at least” $x$ successes.

Core binomial formulas

Exact binomial probability (PMF):

$P$

Binomial coefficient (combinations):

$($

Probability of failure:

$q=1-p$

$P$

Cumulative binomial probabilities

Cumulative “at most” $x$:

$P(X\le x)=\sum _{k=0}^x(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$

Cumulative “at least” $x$:

$P(X\ge x)=\sum _{k=x}^n(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$

Between $m$ and $x$ inclusive:

$P(m\le X\le x)=\sum _{k=m}^x(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$

$q$

​Mean and spread formulas

For a binomial random variable $X\sim \text{Bin}(n,p)$:

Mean (expected value):

$\mu =E[X]=np$

Worked example

Example: A fair coin ($p=0.5$) is flipped $n=5$ times.
Find the probability of exactly $x=3$ heads.

Solution:

$n$

Use the binomial formula:

$P$

Simplify:

$P$

$\mathrm{<br>}(X=3)=(\genfrac{}{}{0ex}{}{5}{3})(0.5{)}^5=10\times \frac{1}{32}=\frac{10}{32}=\frac{5}{16}$

$\frac{\mathrm{<br>}}{}$So the probability of exactly 3 heads in 5 fair coin tosses is $\frac{5}{16}$.

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