AcademyProbability
Academy
Binomial Distribution
Level 1 - Math II (Physics) topic page in Probability.
Principle
A binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same success probability.
Notation
\(Bernoulli trial\)
one trial with two outcomes: success or failure
\(p\)
success probability on each trial
\(1-p\)
failure probability on each trial
\(n\)
fixed number of trials
\(X\)
count of successes in the n trials
\(k\)
a possible success count
\(X\sim\operatorname{Bin}(n,p)\)
X has a binomial distribution with n trials and success probability p
Method
Step 1: Check the trial structure
There must be a fixed number \(n\) of trials, each trial must have success or failure, trials must be independent, and the same \(p\) must apply to every trial.
Step 2: Count success positions
To get exactly \(k\) successes, choose which \(k\) of the \(n\) trial positions are successes.
Choose success positions
\[\binom{n}{k}\]
Probability for one fixed pattern
\[p^k(1-p)^{n-k}\]
Multiply by number of patterns
\[P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\]
Rules
Binomial model
\[X\sim \operatorname{Bin}(n,p)\]
Binomial PMF
\[P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\]
Allowed counts
\[k=0,1,2,\ldots,n\]
Mean
\[E[X]=np\]
Variance
\[\operatorname{Var}(X)=np(1-p)\]
Examples
Question
A fair coin is tossed
\[5\]
times. Find the probability of exactly \[3\]
heads.Answer
Here
\[X\sim\operatorname{Bin}(5,0.5)\]
\[P(X=3)=\binom53(0.5)^3(0.5)^2=10(0.5)^5=\frac{10}{32}.\]
Checks
- The number of trials \(n\) must be fixed before observing outcomes.
- The trials must be independent.
- The same success probability \(p\) must apply to every trial.
- The count must satisfy \(0\le k\le n\).
- The binomial variable counts successes, not the order in which they occurred.