Editing Binomial distribution (section)

=== Probability mass function ===

If the [[random variable]] {{mvar|X}} follows the binomial distribution with parameters {{math|''n'' ∈ [[natural number|<math>\mathbb{N}</math>]]}} and {{math|''p'' ∈ {{closed-closed|0, 1}}}}, we write {{math|''X'' ~ ''B''(''n'', ''p'')}}. The probability of getting exactly {{mvar|k}} successes in {{mvar|n}} independent Bernoulli trials (with the same rate {{mvar|p}}) is given by the [[probability mass function]]:
: <math>f(k,n,p) = \Pr(X = k) = \binom{n}{k}p^k(1-p)^{n-k}</math>
for {{math|''k'' {{=}} 0, 1, 2, ..., ''n''}}, where
: <math>\binom{n}{k} =\frac{n!}{k!(n-k)!}</math>
is the [[binomial coefficient]]. The formula can be understood as follows: {{math|''p''{{sup|''k''}} ''q''{{sup|''n''−''k''}}}} is the probability of obtaining the sequence of {{mvar|n}} independent Bernoulli trials in which {{mvar|k}} trials are "successes" and the remaining {{math|''n'' − ''k''}} trials result in "failure". Since the trials are independent with probabilities remaining constant between them, any sequence of {{mvar|n}} trials with {{mvar|k}} successes (and {{math|''n'' − ''k''}} failures) has the same probability of being achieved (regardless of positions of successes within the sequence). There are <math display="inline">\binom{n}{k}</math> such sequences, since the binomial coefficient <math display="inline">\binom{n}{k}</math> counts the number of ways to choose the positions of the {{mvar|k}} successes among the {{mvar|n}} trials. The binomial distribution is concerned with the probability of obtaining ''any'' of these sequences, meaning the probability of obtaining one of them ({{math|''p''{{sup|''k''}} ''q''{{sup|''n''−''k''}}}}) must be added <math display="inline">\binom{n}{k}</math> times, hence <math display="inline">\Pr(X = k) = \binom{n}{k} p^k (1-p)^{n-k}</math>.

In creating reference tables for binomial distribution probability, usually, the table is filled in up to {{math|''n''/2}} values. This is because for {{math|''k'' > ''n''/2}}, the probability can be calculated by its complement as
: <math>f(k,n,p)=f(n-k,n,1-p). </math>

Looking at the expression {{math|''f''(''k'', ''n'', ''p'')}} as a function of {{mvar|k}}, there is a {{mvar|k}} value that maximizes it. This {{mvar|k}} value can be found by calculating
: <math> \frac{f(k+1,n,p)}{f(k,n,p)}=\frac{(n-k)p}{(k+1)(1-p)} </math>
and comparing it to 1. There is always an integer {{mvar|M}} that satisfies<ref>{{cite book |last=Feller |first=W. |title=An Introduction to Probability Theory and Its Applications |url=https://archive.org/details/introductiontopr01wfel |url-access=limited |year=1968 |publisher=Wiley |location=New York |edition=Third |page=[https://archive.org/details/introductiontopr01wfel/page/n167 151] (theorem in section VI.3) }}</ref>
: <math>(n+1)p-1 \leq M < (n+1)p.</math>

{{math|''f''(''k'', ''n'', ''p'')}} is monotone increasing for {{math|''k'' < ''M''}} and monotone decreasing for {{math|''k'' > ''M''}}, with the exception of the case where {{math|(''n'' + 1)''p''}} is an integer. In this case, there are two values for which {{mvar|f}} is maximal: {{math|(''n'' + 1) ''p''}} and {{math|(''n'' + 1) ''p'' − 1}}. {{mvar|M}} is the ''most probable'' outcome (that is, the most likely, although this can still be unlikely overall) of the Bernoulli trials and is called the [[Mode (statistics)|mode]].

Equivalently, {{math|''M'' − ''p'' < ''np'' ≤ ''M'' + 1 − ''p''}}. Taking the [[Floor and ceiling functions|floor function]], we obtain {{math|''M'' {{=}} floor(''np'')}}.{{NoteTag|Except the trivial case {{math|''p'' {{=}} 0}}, which must be checked separately.}}