# What is Binomial: Definition and 667 Discussions

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution remains a good approximation, and is widely used.

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1. ### B I've been trying to understand the proof for the binomial theorem

Hello everyone, I've been trying to understand the proof for the binomial theorem and have been using this inductive proof for understanding. So far the proof seems consistent everywhere it's explicit with the pattern it states, but I've started wondering if I actually fully grock it because I...
2. ### A How Can We Analyze an Exam with Varying Multiple Choice Options?

If we had a multiple choice exam with , say, 20 questions, with 4 choices for each question, we can analyze it as a Binomial(20, .25). What if instead , some of the questions offered 2,3, 4, etc., choices? Is there a " nice" way of analyzing the exam as a whole?
3. ### Find the sum of the coefficients in the expansion ##(1+x)^n##

##(1+x)^n=1+C_1x+C_2x^2+C_3x^3...+C_nx^n## Let ##x=1##, hence ##2^n=1+C_1+C_2+C_3...+C_n## which is equal to the sum of the coefficients. I originally thought the sum of the coefficients would be ##2^n-1## since the very first term ##1## is just a number and has no variable. But apparently...
4. ### POTW Does the Alternating Binomial Sum Formula Hold for All Positive Integers?

Show that for all positive integers ##n##, $$\binom{n}{1} - \frac{1}{2}\binom{n}{2} + \cdots + (-1)^{n-1}\frac{1}{n}\binom{n}{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$$
5. ### A Is the binomial a special case of the beta binomial?

On Wikipedia one can read in the article Beta-binomial distribution: > It also approximates the binomial distribution arbitrarily well for > large ##\alpha## and ##\beta##. where 'It' refers to the beta-binomial distribution. What does 'arbitrarily well' mean here?
6. ### Use binomial theorem to find the complex number

This is also pretty easy, ##z^5=(a+bi)^5## ##(a+bi)^5= a^5+\dfrac {5a^4bi}{1!}+\dfrac {20a^3(bi)^2}{2!}+\dfrac {60a^2(bi)^3}{3!}+\dfrac {120a(bi)^4}{4!}+\dfrac {120(bi)^5}{5!}## ##(a+bi)^5=a^5+5a^4bi-10a^3b^2-10a^2b^3i+5ab^4+b^5i## ##\bigl(\Re (z))=a^5-10a^3b^2+5ab^4## ##\bigl(\Im (z))=...

46. ### MHB Determine an expression using binomial theorem

Determine an expression for f(x) =(1+x)(1+2x)(1+3x)…(1 +nx),find f⸍(0) .
47. ### Am I justified in using the binomial distribution?

Homework Statement 12 non-distinguishable attacks from President Snow land in Panem’s 12 districts in a particular week. Assume the attacks are located randomly, with each configuration of attacks equally likely. What is the probability that some district had more than 1 attack? Homework...
48. ### Combinations possible when choosing 4 or 5 team members from

Homework Statement How many combinations of people are there if you choose 4 or 5 from a group of 10? Homework Equations Relies on binomials The Attempt at a Solution binomial (10,4) = binomial (10,6) = 210 But when choosing 5 the answer is binomial (10,5) / 2 = 126 Why do I need to divide by 2?
49. ### Binomial Expansion (Arfken/Weber/Harris 1.3.9)

Hi everyone, I'm currently working through Mathematical Methods for Physicists 7th ed. by Arfken/Weber/Harris and there's one question that's been giving me some difficulty. I would appreciate any feedback if possible. Thanks! Chris Homework Statement The relativistic sum w of two...
50. ### Derivation of Bernoulli Binomial distribution

Homework Statement Derive the bernoulli binomial distribution.Homework EquationsThe Attempt at a Solution Each bernoulii trial could have only two possible outcomes . Let’s name one outcome as success and another outcome as failure. Let’s denote the probability of getting success and failure in...