## Binomial Distribution

*Binomial distribution* is the distribution of a total number of successes in a given number of Bernoulli trials. The common notation is ^{k}(1 - p)^{n - k}.

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For a fixed number of trials n, the binomial distribution always behaves in the same way: as a function of k, it is monotone increasing up to a certain point m after which (perhaps with an exception of the next point) it is monotone decreasing.

Indeed,

b(k; n, p) / b(k-1; n, p) | = C(n, k) p^{k}(1 - p)^{n - k} / C(n, k-1) p^{k-1}(1 - p)^{n - k + 1} | |

(1) | = (n - k + 1) p / k(1 - p) | |

= 1 + ((n + 1) p - k) / k(1 - p). | ||

It is clear now that the right hand side in (1) is greater than 1 whenever

b(m; n, p) = b(m - 1; n, p). | |

In any event, there is only one integer m that satisfies

(2) | (n + 1) p - 1 < m ≤ (n + 1) p. |

Summing up, as a function of k, the expression b(k; n, p) is monotone increasing for

The number m that satisfies (2) is known as the *most probable* (*most likely*) number of successes in n Bernoulli trials. As a matter of fact,

∑ k b(k; n, p) | = ∑ k C(n, k) p^{k}q^{n - k} | |

= pq^{n-1}∑ k C(n, k) (p/q)^{k - 1} | ||

= pq^{n-1} n (1 + p/q)^{n - 1} | ||

= pq^{n-1} n (q + p)^{n - 1} / q^{n - 1} | ||

= p n, | ||

where we used the identity

n (1 + x)^{n - 1} | = ∑ k C(n, k) x^{k - 1}. | |

The result for the expected value np = ∑ k b(k; n, p) might have been anticipated given the interpretation of the probability as a relative frequency.

Note that the expected value np is always different from the most likely value

- What Is Probability?
- Intuitive Probability
- Probability Problems
- Sample Spaces and Random Variables
- Probabilities
- Example: A Poker Hand
- Bernoulli Trials
- Binomial Distribution
- Proofreading Example

- Conditional Probability
- Dependent and Independent Events
- Algebra of Random Variables
- Expectation
- Probability Generating Functions
- Probability of Two Integers Being Coprime
- Random Walks
- Probabilistic Method
- Probability Paradoxes
- Symmetry Principle in Probability
- Non-transitive Dice

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