Binomial probability distribution table
Let \(X\) be the number of successes in \(n\) independent trials, where each trial has success probability \(p\). Then \(X\) follows a binomial distribution: \(X \sim \text{Bin}(n,p)\). A binomial probability distribution table provides precomputed probabilities so that values like \(P(X \le x)\) can be read without repeated calculation.
What the table typically contains
Many textbooks tabulate cumulative probabilities \(P(X \le x)\) for selected \(n\), \(p\), and integer \(x\). Some tables instead list exact probabilities \(P(X=x)\). The conversion rules below work for either format once the table’s definition is identified.
How to use a cumulative binomial table \(P(X \le x)\)
- Identify the model \(X \sim \text{Bin}(n,p)\) and confirm the value of \(n\) and \(p\).
- Locate the section for \(n\), then the column (or subcolumn) for \(p\), and finally the row for \(x\).
- Read the entry \(F(x)=P(X \le x)\) from the table.
- Convert to the desired probability using complements or differences.
Common probability requests and table conversions
| Goal | How to get it from a cumulative table \(F(x)=P(X\le x)\) |
|---|---|
| \(P(X\le x)\) | Read directly: \(F(x)\). |
| \(P(X< x)\) | \(P(X<x)=F(x-1)\) (for integer \(x\)). |
| \(P(X = x)\) | \(P(X=x)=F(x)-F(x-1)\), with \(F(-1)=0\). |
| \(P(X\ge x)\) | \(P(X\ge x)=1-F(x-1)\). |
| \(P(a \le X \le b)\) | \(P(a\le X\le b)=F(b)-F(a-1)\). |
| \(P(a < X < b)\) | \(P(a<X<b)=F(b-1)-F(a)\). |
Worked example using a binomial probability distribution table
Suppose a table lists cumulative probabilities for \(X \sim \text{Bin}(8,0.5)\). The excerpt below shows both \(P(X=x)\) and \(P(X\le x)\) for clarity (a textbook table may show only one of these columns).
| \(x\) | \(P(X=x)\) | \(P(X\le x)\) |
|---|---|---|
| 0 | 0.0039 | 0.0039 |
| 1 | 0.0312 | 0.0352 |
| 2 | 0.1094 | 0.1445 |
| 3 | 0.2188 | 0.3633 |
| 4 | 0.2734 | 0.6367 |
| 5 | 0.2188 | 0.8555 |
| 6 | 0.1094 | 0.9648 |
| 7 | 0.0312 | 0.9961 |
| 8 | 0.0039 | 1.0000 |
Example A: Find \(P(X\le 2)\).
\[ P(X\le 2)=F(2)=0.1445. \]
Example B: Find \(P(X=3)\) using only cumulative values.
\[ P(X=3)=F(3)-F(2)=0.3633-0.1445=0.2188. \]
Example C: Find \(P(X\ge 6)\).
\[ P(X\ge 6)=1-F(5)=1-0.8555=0.1445. \]
Example D: Find \(P(2\le X\le 5)\).
\[ P(2\le X\le 5)=F(5)-F(1)=0.8555-0.0352=0.8203. \]
When the table’s \(p\) range is limited
Some binomial tables only list values for \(p \le 0.5\). If \(p>0.5\), set \(q=1-p\) and define \(Y=n-X\). Then \(Y \sim \text{Bin}(n,q)\), and events in \(X\) can be rewritten in terms of \(Y\). For example:
\[ P(X\le x)=P(n-X \ge n-x)=P(Y \ge n-x)=1-P(Y\le n-x-1). \]