Binomial probabilities may be obtained using ...

Excel

In Excel, the function "BINOM.DIST()" can be used to find binomial probabilities. For example, if \(X \sim \BinomDistn(n=20, \pi=0.3)\),

\(\mathbf {P(X = 4)}\)
Type in an Excel spreadsheet cell "=BINOM.DIST(4, 20, 0.3, false)"
\(\mathbf {P(X ≤ 2)}\)
Type in an Excel spreadsheet cell "=BINOM.DIST(2, 20, 0.3, true)"

Scientific calculator

If \(X \sim \BinomDistn(n=20, \pi=0.3)\) then a scientific calculator can find:

\(\mathbf {P(X = 4)}\)
\[ p(4)= {20 \choose 4} {0.3}^4(1-0.3)^{20-4} \]
\(\mathbf {P(X ≤ 2)}\)
\[ \begin {align} p(0) + p(1) + p(2) &= {20 \choose 0} {0.3}^{0}0.7^{20} + {20 \choose 1} {0.3}^{1}0.7^{19} + {20 \choose 2} {0.3}^{2}0.7^{18} \\ & =\;\; 0.7^{20} \;+\; 20 \times 0.3 \times 0.7^{19} \;+\; 190 \times {0.3}^{2}0.7^{18} \end {align}\]

Probabilities for ranges of counts

Care must be taken with the wording of questions — should the 'extreme' value that is mentioned in the wording of the interval be included? For example,

In words...     In terms of X         Using ½    
More than 5 X > 5 X > 5.5
Greater than or equal to 5 X ≥ 5 X > 4.5
No more than 5 X ≤ 5 X < 5.5
At least 5 X ≥ 5 X > 4.5
Fewer than 5 X < 5 X < 4.5
5 or fewer X ≤ 5 X < 5.5

The final column most clearly expresses which values of \(X\) are included.

We recommend translating any interval into its form using ½ before finding its probability.

Translating the interval in this way is particularly useful when using a normal approximation to evaluate the probability. In this context, it is called a continuity correction.