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. C

    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. W

    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. RChristenk

    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. Euge

    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

    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. chwala

    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))=...
  7. chwala

    Solve the equation involving binomial theorem

    $$(7-6x)^3+(7+6x)^3=1736$$ $$⇒(7^3(1-\frac {6}{7}x)^3+(7^3(1+\frac {6}{7}x)^3=1736$$ $$343[1-\frac {18}{7}x+\frac {216}{98}x^2-\frac{1296}{2058}x^3]+343[1+\frac {18}{7}x+\frac {216}{98}x^2+\frac{1296}{2058}x^3]=1736$$ $$343[2+\frac {432}{98}x^2]=1736$$ $$686+\frac {148,176}{98}x^2=1736$$ $$\frac...
  8. L

    MHB What is the Minimum Number of Friends Needed for Unique Dinner Invitations?

    Hello all, I am trying to solve this one: John has n friends . He wants to invite in each evening (365 days a year) three of his friends for dinner. What should be the size of n, such that it will be possible not to invite the same triplet twice ? What I did was: \[\binom{n}{3}\leq 365\]...
  9. S

    B Continuity correction when using normal as approximation for binomial

    What if the value of X is not integer, such as P(X < 1.2)? a) Will the continuity correction be P(X < 1.2 - 0.5) = P(X < 0.7)? or b) Will the continuity correction be P(X < 1.2 - 0.05) = P(X < 1.15)? or c) Something else? Thanks
  10. Ackbach

    MHB Binomial Distribution: Likelihood Ratio Test for Equality of Several Proportions

    $\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: A survey of voter sentiment was conducted in four midcity political wards to compare the fraction of voters favoring candidate $A.$ Random samples of $200$ voters were polled in each of the...
  11. C

    MHB Identifying if an experiment is a binomial experiment

    There is an example : A box contains 20 cell phones, and two of them are defective. Three cell phones are randomly selected from this box and inspected to determine whether each of them is good or defective. Is this experiment a binomial experiment? AND the answer is : NOT a binomial experiment...
  12. A

    I Proving the Finite Binomial Series for k Non-Negative Integer

    Hello, I was wondering how to prove that the Binomial Series is not infinite when k is a non-negative integer. I really don't understand how we can prove this. Do you have any examples that can show that there is a finite number when k is a non-negative integer? Thank you!
  13. AllThatJaz22

    MHB How Many Iterations Can Be Expected from Drawing Cards in Better Backstories?

    Howdy! I have a deck of cards I created called Better Backstories. The Basic Deck is made up of 60 cards and each one has a unique title. 38 of the cards have a chart of 10 suggestions, and the remaining 22 have flavor text that could reasonably include 3 suggestions. So, the total number of...
  14. issue

    Important help on the subject of polynomials of binomial arrangement

    [Mentor Note -- Multiple threads merged. @issue -- please do not cross-post your threads] Hi, everyone It is known that binomial distribution can also be solved by polynomials. i add document with a question I can not solve. Glad to get for help Thanks to all the respondents
  15. issue

    Important help on the subject of polynomials of binomial arrangement

    Hi, everyone It is known that binomial distribution can also be solved by polynomials. i add document with a question I can not solve. Glad to get for help Thanks to all the respondents
  16. S

    Find the middle terms of this binomial expansion

    I know how to expand it. My question is: the expansion has 8 terms so what would be the middle term? Will the answer be "the expansion has no middle term"? Or maybe seeing the phrase of the question (middle terms), there will be two answers (the 4th and 5th term)? Thanks
  17. entropy1

    I Binomial distribution of "worlds" in MWI

    If we have a spin measurement with P(up)=0.5 en P(down)=0.5, this is equivalent to tossing a coin P(heads)=0.5 and P(tails)=0.5. The probability of having five heads and five tails out of ten tosses is the binomial: ##\binom{10}{5}(0.5)^5(0.5)^5##. So the same would hold for the spin...
  18. AN630078

    Binomial Expansion: Evaluating Coefficient from two binomials

    So I think I may be overcomplicating this problem but I realize that in order to find the x^3 term it will be the product of the two binomials, ie. x^1*x^2=x^3. The coefficient of x^3 will be the coefficient of x^1 in the first bracket multiplied by the coefficient of x^2 in the second bracket...
  19. M

    MHB Digital Clocks: Binomial Problem Analysis and Results

    A company makes digital clocks. It is determined that 5% of all clocks produced are defective. you go to the warehouse and randomly select 80 clocks. 1. How many of the 80 clocks do you expect to be defective? 2.What is the probability that exactly 6 of the clocks are defective? 3. What is...
  20. S

    B Binomial Theorem: Exploring Meaning of Coefficients in General Expansion

    In the general expansion of (1+x)^n what does the sum of the coefficients mean?
  21. L

    Square-Root of a binomial squared

    Suppose that ##a##, ##b##, and ##x## are integers. How would the ##±##s be correctly assigned in such an equation?
  22. CaptainX

    B Coin Tossing: Binomial Distribution Explained

    Why tossing a coin three times is said to have binomial distribution? I'm little bit confused.
  23. benorin

    B Does the binomial distribution play a role determining p from data?

    In a game heroes have a maximum dodge rate, from experimental data we have 13 dodges out of 24 attacks (so 11 hits). A fellow on my discord server had immediately solved for the dodge rate as being 13/24. I started to explain it is not so simple as dividing (24-11)/24=13/24 is not the dodge...
  24. Biochemgirl2002

    Approximation to the binomial distrubution

    a) since np has to be greater than 5, n*p= 50*.5 =25 so yes, we can use this since it is much larger than 5. now, for mean, i believe the equation is saying that the mean is np, which is 25 but in this equation we do not have a q value, so this is where my issue begins... what should i use...
  25. benorin

    B How to handle probabilities of the number of trials in a Binomial distribution

    Suppose our process has a 85% chance of 2 trials and a 15% chance of 3 trials, and the rest is straightforward binomial distribution, do I take the weighted average of the binomial distribution at 2 and 3 trials? This is for a game so, yeah thanks.
  26. CrosisBH

    I How is a binomial expansion done?

    Summary: Can someone give me a basic high level overview on how to do a binomial expansion? I'm studying for my E&M test and going over multipole expansion. I'm particularly confused about these lines (Griffiths E&M 4th Edition) 𝓇^2_{\pm} = r^2 \left(1\mp \frac{d}{r} \cos\theta +...
  27. S

    I Generating samples on a 2-D composite binomial distribution

    I would like to generate (X,Y) pairs such that they would follow a distribution something like this: This is the sum of three normal distributions. Each distribution could have a different taper along the X and the Y, plus an offset along X and/or Y. So the parameters of these three...
  28. B

    MHB Please help for binomial expansion (2x-1/(2x^2))^9

    As titled, been cracking my head over it. Thanks in advance!
  29. opus

    Maple Help with a Maple Program: Binomial Coefficients

    Please see attached image. I'm not sure why I'm getting this error because this is the format I have used to write programs in Maple before. Any ideas? I'm new to this so not sure how to independently trouble shoot or problem solve this, Thanks!
  30. D

    I Binomial theorem with more than 2 terms

    Hi. Is the binomial theorem ##(1+x)^n = 1+nx+(n(n-1)/2)x^2 + ….## valid for x replaced by an infinite series such as ##x+x^2+x^3+...## with every x in the formula replaced by the infinite series ? If so , does the modulus of the infinite series have to be less than one for the series to...
  31. R

    I'm having difficulty expressing a binomial expansion as a sum

    I found the first 4 terms of the series: ½-(1/16)x^2+(1/64)x^4-(7/1536)x^6. I cannot however simplify this to a sum. the 7 in the numerator of the last term of the above expansion is the sticking point.
  32. S

    I Sum of Binomial Expansion | Spivak Chapter 2, Excercise 3 part d

    Hello, I am working through Spivak for self study and sharpening my math skills. I have become stuck on an exercise. What I need to show is the following: $$ (a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = \sum_{j = 0}^{n + 1} \binom{n+1}{j} a^{n-j + 1}b^{j} $$ My attempt, starting from...
  33. Eclair_de_XII

    How to prove the binomial coefficient?

    Basically, the way I did this problem was to use a table with a known ##n## and ##k##. In this case, I fixed ##n=5##, and ##k=3##. I wanted to find the number of terms with three ##x##'s and two ##y##'s. I labeled each ##x_i##, ##1\leq i \leq 5##; the ##y_i## are labeled the same way. Anyway...
  34. lfdahl

    MHB Prove the binomial identity ∑(-1)^j(n choose j)=0

    Prove the binomial identity: $$\sum_{j=0}^{n}(-1)^j{n \choose j}=0$$ - in two different ways
  35. G

    A Binomial as a sum of tetranomials

    Hello there, I'm working on a kinetic theory of mixing between two species - b and w. Now, if I want to calculate the number of different species B bs and W ws can form, I can use a simple combination: (W+B)!/(W!B!) Now, in reality in my system, ws and bs form dimers - ww, bb, wb and bw...
  36. J

    MHB Binomial Sum \displaystyle \sum^{n}_{k=0}\binom{n+k}{k}\cdot \frac{1}{2^k}

    Evaluation of $\displaystyle \sum^{n}_{k=0}\binom{n+k}{k}\cdot \frac{1}{2^k}$
  37. YoungPhysicist

    B Understanding the Evolution of Binomial Coefficient Notation: Old vs. New

    I am learning binomial theorem now on my long journey to calculus. I noticed that in older textbooks, the binomial coefficient looks like C(n on top,k on bottom) I don’t think that I can display it here and in newer ones,they look like ##\binom{n}{k}## is the old notation outdated?or this is...
  38. J

    Using binomial coefficients to find sum of roots

    Homework Statement >Find the sum of the roots, real and non-real, of the equation x^{2001}+\left(\frac 12-x\right)^{2001}=0, given that there are no multiple roots. While trying to solve the above problem (AIME 2001, Problem 3), I came across three solutions on...
  39. backtoschool93

    Variance of binomial distribution

    Homework Statement Random variable Y has a binomial distribution with n trials and success probability X, where n is a given constant and X is a random variable with uniform (0,1) distribution. What is Var[Y]? Homework Equations E[Y] = np Var(Y) = np(1-p) for variance of a binomial...
  40. H

    Expected value of binomial distribution

    Homework Statement A random variable Y has a binomial distribution with n trials and success probability X, where n is a given constant and X is a uniform(0,1) random variable. What is E[Y]? Homework Equations E[Y] = np The Attempt at a Solution The key is determining the probability of...
  41. Z

    Bernoulli, Binomial & Poisson: What is pi?

    Homework Statement Hi, I have a confusion in knowing Pi in the equations attached. Eq are related to the Topic Discrete Random Variables in the context of Probability lecture I also can't understand what is P(Y=y|Pi)? Homework Equations Eq are attached The Attempt at a Solution I can't...
  42. A

    I Accuracy of the Normal Approximation to Binomial

    What is the preferred method of measuring how accurate the normal approximation to the binomial distribution is? I know that the rule of thumb is that the expected number of successes and failures should both be >5 for the approximation to be adequate. But what is a useful definition of...
  43. chwala

    Finding the convergence of a binomial expansion

    Homework Statement Expand ##(1+3x-4x^2)^{0.5}/(1-2x)^2## find its convergence valueHomework EquationsThe Attempt at a Solution on expansion ##(1+3/2x-3.125x^2+4.6875x^3+...)(1+4x+12x^2+32x^3+...)## ##1+5.5x+14.875x^2+42.1875x^3+... ## how do i prove for convergence here?
  44. Sarina3003

    Generating functions, binomial coefficients

    Homework Statement a) I have to find and expression for sequence of $b_n$ in terms of generating functions of the sequence of $a_n$ $$b_n = (-1)^{n}(n+1)a_0 +(-1)^{n-1}n a_1+...+(-1)2a_{n-1}+a_n$$ with $$a_n = a_{n-1} +8a_{n-2} -12a_{n-3} +25(-3)^{n-2} + 32n^2 -64$$ b) I have to use the...
  45. E

    MHB Determine an expression using binomial theorem

    Determine an expression for f(x) =(1+x)(1+2x)(1+3x)…(1 +nx),find f⸍(0) .
  46. K

    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...
  47. Z

    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?
  48. CJ2116

    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...
  49. Pushoam

    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...