Distributions

Binomial Probability

A binomial probability counts the ways to get k successes and multiplies by the probability of each success/failure pattern:

P(X=k)=C(n,k)·p^k·(1-p)^(n-k)
binomial-probability P(X=k)=C(n,k)·p^k·(1-p)^(n-k)

Example

Compute the combination factor and probability powers.

highlighted = computed this step

Step 1 — Parameters

List n, k, and p.

n=4k=2p=12n= \hl{4} \quad k= \hl{2} \quad p= \hlmath{\frac{1}{2}}

Step 2 — Combinations

Compute the combination factor.

C(4,2)=6C( 4 , 2 )= \hl{6}

Step 3 — Power of p

Compute the power of p.

pk=(12)2=14p^k=( \frac{1}{2} )^{ 2 }= \hlmath{\frac{1}{4}}

Step 4 — Power of one minus p

Compute the power of one minus p.

q=1−12=12qn−k=(12)2=14q= 1 - \frac{1}{2} = \frac{1}{2} \quad q^{n-k}=( \frac{1}{2} )^{ 2 }= \hlmath{\frac{1}{4}}

Step 5 — Multiply factors

Multiply the binomial factors.

P(X=k)=6⋅14⋅14=38P(X=k)= 6 \cdot \frac{1}{4} \cdot \frac{1}{4} = \hlmath{\frac{3}{8}}