Binomial Definition and 668 Threads

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. P

    Binomial theorem proof by induction

    On my problem sheet I got asked to prove: ## (1+x)^n = \displaystyle\sum _{k=0} ^n \binom{n}{k} x^k ## here is my attempt by induction... n = 0 LHS## (1+x)^0 = 1 ## RHS:## \displaystyle \sum_{k=0} ^0 \binom{0}{k} x^k = \binom{0}{0}x^0 = 1\times 1 = 1 ## LHS = RHS hence true for...
  2. paulmdrdo1

    MHB Find Term with $x^2$ in Binomial Theorem

    find the term with $x^2$ $\displaystyle\left(x^2-\frac{1}{x}\right)^{10}$ thanks!
  3. applestrudle

    Binomial theorem to evaluate limits?

    Homework Statement lim x->1 (X^9 + x -2)/(x^4 + x -2) I know how to do this using L'Hopitals Rule and I get 2 Homework Equations (1+b)^n = 1 + bn + n(n-1)b^2/2! + n(n-1)(n-2)b^3/3! ... The Attempt at a Solution Let x = h+1 x -> 1 h -> 0 lim h->0 (h+1)^9 +...
  4. J

    Probability: binomial coefficient problem

    Hello physics forum! I come to you with a binomial coefficient problem I am stuck on. Here is the question 1. Suppose an airport has three restaurants open, Subway Burger King and McDonalds. If all three restaurants are open and each customer is equally likely to go to each one, what is the...
  5. D

    Binomial expansion of a function with x raised to a power

    Hey guys. So I need to know how to Binomial expand the following function \frac{1}{(1-x^{2})}. I need this because I have to work out \prod^{∞}_{i=1}\frac{1}{(1-x^{i})} for i up to 6. But I have to do it with Binomial expansion. If i can learn how to do \frac{1}{(1-x^{2})} then the rest...
  6. MarkFL

    MHB Apply Binomial Theorem: Expand (x-2y)^3

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  7. hxthanh

    MHB What is the result of the sum of binomial coefficients with alternating signs?

    Evaluate sum: $\displaystyle S=\sum_{k=0}^{2n}(-1)^k{2n\choose k}{4n\choose 2k}$
  8. R

    Can Binomial Distribution Be Approximated to Poisson Distribution?

    Homework Statement The question requires me to approximate binomial distribution to get poisson distribution. Show that N!/(N-n)!=N^n. Homework Equations N!/n!(N-n)! p^n q^(N-n)=Binomial distribution The Attempt at a Solution I expanded N!/(N-n)! and got...
  9. M

    What is the Probability of Selling 8 Listings Out of 10?

    Homework Statement You are a real estate salesperson and you currently have 10 listings. Past experience has shown that you will sell approximately 70% of your listings. If sales are independent: What is the probability that you make exactly 8 sales? Homework Equations The Attempt...
  10. polygamma

    MHB A sum involving the central binomial coefficients

    Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Big[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Big]- \frac{4}{3} \zeta(3) $$ where $\psi_{1}(x)$ is the trigamma function. But I can't get my answer in...
  11. Seydlitz

    Proving a formula with binomial coefficient when n=-1

    Homework Statement Prove that ##\binom{-1}{k}=(-1)^k## The Attempt at a Solution Using induction on ##k##, ##\binom{-1}{0}=1## which is true also for ##(-1)^0=1## Assuming ##\binom{-1}{k}=(-1)^k##, then ##\binom{-1}{k+1}=(-1)^{k+1}## Indeed when ##n=-1##, we can write...
  12. T

    Interpretation for identity with binomial coefficients

    I am looking for a counting interpretation to make the following identity evident: \sum_{k=0}^{n-j}(-1)^k\binom{j-1+k}{j-1}\binom{n}{j+k} = 1 The form of it looks like inclusion-exclusion. The sum is 1, more or less independent of j. So that makes me think it would be something like "how...
  13. S

    MHB How do I factor a binomial with a coefficient of 4?

    I'm having trouble factoring the following binomial... can someone try to point me in the right direction please? 4y^3+4 It has been 7 years since I took algebra and I am trying to get my review done. This one just does not make sense to me right now... Thanks!
  14. F

    Binomial formula for spherical tensors

    We know that the Newton binomial formula is valid for numbers in elementary algebra. Is there an equivalent formula for commuting spherical tensors? If so, how is it? To be specific let us suppose that A and B are two spherical tensors of rank 1 and I want to calculate (A + B)4 and I want...
  15. Q

    Can Binomial Distribution Be Used for Small Populations?

    Hello all! I'm trying to understand whether I can use the binomial distribution in a certain way... According to the equation, to find the probability P of a certain number of successes out of a number you trials, you need the number of trials, n; the number of successes out of the trials...
  16. M

    What is the solution to the Binomial Theorem problem highlighted in red?

    I highlighted the portion in red in the paint document that I'm not understanding. How can we see by inspection that the product is equal to the series 2?
  17. A

    Predicting Absenteeism: Comparing Binomial Distribution in Two Classes

    Homework Statement In a class with 20 and one with 10 students each student has a probability of 0.3 to not show up on a particular day. On a given day, which class is most likely to have the highest ratio of absent students? This was in my exam, unfortunately I did not know how to do it...
  18. reenmachine

    Finding the Coefficient of x^6y^3 using Binomial Theorem in (3x-2y)^9

    Homework Statement Use the binomial theorem to find the coefficient of ##x^6y^3## in ##(3x-2y)^9##. Homework Equations ##1+9+36+84+126+126+84+36+9+1## (I used two lines for the lenght) ##1(3x)^9(-2y)^0+9(3x)^8(-2y)^1+36(3x)^7(-2y)^2+84(3x)^6(-2y)^3+126(3x)^5(-2y)^4##...
  19. reenmachine

    Finding Coefficient of x^8y^5 using Binomial Theorem

    Homework Statement Use the binomial theorem to find the coefficient of ##x^8y^5## in ##(x+y)^{13}##. Homework Equations We know 13 - 5 = 8 , so we have ##\binom{n}{5}x^{n-5}y^5 = \binom{13}{5}x^8y^5## ##\binom{13}{5} = \frac{13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8!}{5!8!} = \frac{13...
  20. reenmachine

    Binomial Coefficient - Factorials Part III

    Homework Statement ##| \ X \in \mathcal P(\{0,1,2,3,4,5,6,7,8,9\}) : |X|= 4 \ | = \ \ ?## Homework Equations There's no wording in the exercise , just what I wrote above.If I understood correctly , they asked me to find the cardinality of the set of all subsets of {0,1,2,3,4,5,6,7,8,9} that...
  21. reenmachine

    Binomial Coefficient - Factorials Part II

    Homework Statement This one is trickier than the problem in my other thread in my opinion.Twenty-one people are to be divided into two teams , The Red Team and the Blue Team.There will be 10 people on Red Team and 11 people on Blue Team.How many ways to do this? I am not sure how to solve...
  22. reenmachine

    Set Theory - Counting - Binomial Coefficient - Factorials

    Homework Statement A department consists of 5 men and 7 women.From this department you select a committee with 3 men and 2 women.In how many ways can you do this? Homework Equations Since the "overall set" (the entire department) is composed of both men and women and each has a specific...
  23. P

    Binomial Distribution satisfies Marcoff Chain

    1. The problem statement Consider the Binomial Distribution in the form P_{N}(m)=\frac{N!}{(\frac{N+m}{2})!(\frac{N-m}{2})!}p^{\frac{N+m}{2}}q^{\frac{N-m}{2}} where p+q=1, m is the independent variable and N is a parameter. Show that it satisfies the marcoff chain...
  24. E

    Equivalent form for Binomial Expression?

    Is there a way to express ##{n\choose k}{n\choose r}## in another form without n as the "top" of the binomial coefficient? I remember seeing it once but I forgot what it was.
  25. S

    What is the Value of n in a Binomial Theorem Problem?

    1.Find n, if the term 11 coefficient it is 6 time the term 10 coefficient in 2.(6x^7 + 5x^(-4))^n 3
  26. Fernando Revilla

    MHB Binomial series (radius of convergence)

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  27. Y

    Using binomial theorem in exponential

    In page 11 of http://math.arizona.edu/~zakharov/BesselFunctions.pdf, I am trying to follow the derivation using binomial theorem to get this step: (e^{j\theta}-e^{-j\theta})^{n+2k}≈\frac{(n+2K)!}{k!(n+k)!}(e^{j\theta})^{n+k}(-e^{-j\theta})^kIf you read the paragraph right above this equation...
  28. J

    Normal and binomial distribution: using Z-scores to find answer

    The prices for bananes that a fruit shop would have to pay to keep them in stock have a mean of $1.35/kg and a standard deviation of 18 cents. The owner will not pay more than a certain price, but manages to keep stock 8% of the time. What is the maximum price the ownwer will pay? I found...
  29. D

    Help with proof for binomial distribution

    Greetings to you, Physics Forums regulars! Please allow me to introduce myself a bit first. I'm a student in the Life Sciences, so I don't really have a lot of knowledge on mathematics past the basics. I'm not sure if my problem belongs here. This is my first visit to this...
  30. Fernando Revilla

    MHB Binomial theorem (Milind Charakborty's question at Yahoo Answers)

    Here is the question: Here is a link to the question: What are the last three and four terms of (a + b)^n? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  31. L

    Test Hypotheses with sample of Binomial RV's

    Hi, I am trying to teach myself how to test hypotheses for any distribution, but am having some trouble. X=number chosen each year θ=Mean number chosen in the population H0: θ=.5 h1: θ>.5 The random sample of n=4 is 0,1,3,3 Test the Hypotheses at α≤0.05 assuming X is a...
  32. M

    Is a Binomial Distribution the Correct Approach for a Random Walk Problem?

    Random walk or binomial?? Statement: A drunk person wonders aimlessly along a path by going forward 1 step and backward 1 step with equal probabilities of ½. After 10 steps, a) what is the probability that he has moved 2 steps forward? b) What is the probability that he will make it to his...
  33. L

    Testing Hypotheses for Binomial Distributions: A Beginner's Guide

    Hi, I am trying to teach myself how to test hypotheses for any distribution, but am having some trouble. X=number chosen each year θ=Mean number chosen in the population H0: θ=.5 h1: θ>.5 The random sample of n=4 is 0,1,3,3 Test the Hypotheses at α≤0.05 assuming X is a...
  34. J

    How Likely Are Specific Answer Distributions on Multiple Choice Exams?

    Hi, I am new here, and my name is Jonas. I'm a CS major at a university in the Northeast US. I'm a senior and wrapping up degree requirements which include a science track. I chose Chemistry because Physics was full. The chemistry exams are multiple choice (because you couldn't grade 300...
  35. S

    Binomial theorem induction proof

    Homework Statement Will someone be kind enough to check my proof (attached) of the following (also attached) theorem?Homework Equations The Attempt at a Solution Oh, and as you might notice, I was beginning to run out of paper, but the binomial coefficients in the bracketed terms obviously...
  36. A

    The binomial series coefficient

    Homework Statement Use the binomial series to expand the function as a power series. 1/(2+x)3 I have attached an image. I understand until the end of the second line. I don't see the reasoning used to follow through to the third line. the (-1)^n is because the sign alternates becoming...
  37. T

    Binomial Coefficient Equivalency

    Find an expression that is identical to \sum_{k=0}^n \binom{3n}{3k} According to Wolfram, the correct solution to this is: \frac{1}{3} \left(2(-1)^n + 8^n\right) But I'm not sure which identities of the binomial coefficient I'm supposed to use to prove this. Can anyone give me some...
  38. Petrus

    MHB Constant term in a binomial expansion

    Decide constant term in \left(3\cdot x^3+\left(\frac{-4}{x} \right) \right)^{20}. I have problem with this one, I can't find any example about this one in my book, any advice would be great:)
  39. trollcast

    Could you use the binomial distribution here?

    I'm looking through my statistics notes and on the page that's giving examples of cases where you can use a binomial distribution it gives the problem: "The number of red counters in a randomly chosen sample of 30 counters taken from a large number of counters of which 10% are red." Now...
  40. B

    Combinatorial question: permutation, binomial coefficient

    How many numbers of 6 digits which have exatctly the digit 1 (2 times), digit 2 (2 times), without zero, are there? The book post this solution: \frac{6!}{2!2!}*\binom{7}{2} + \frac{6!}{2!2!2!}*7= 4410, but I'm trying to find an explanation for this result.
  41. D

    Statistics using binomial expansion

    Homework Statement Uploaded Homework Equations P(y)=(n c y)p^y(q^(n-y)) this is also uploaded The Attempt at a Solution My attempt at the first two parts is uploaded I am really confused on how to do game 6 and 7. Also I am a bit confused on how what I did worked in P(y=5), as the...
  42. D

    Is the Probability of Getting an Even Number of Heads 1/2 After 491 Coin Tosses?

    Homework Statement A fair coin is tossed 491 times. The total number of heads or tails is then even or uneven. Is the probability that the head will result in an even result equal to 1/2 Motivate your answer with a strict mathematical proof. Homework Equations I am having some trouble...
  43. M

    Can Calculating Cumulative Binomial Probabilities Be Simplified?

    Homework Statement Of all the weld failures in a certain assembly, 85% of them occur in the weld metal itself, and the remaining 15% occur in the base metal. A sample of 20 weld failures is examined. a. What is the probability that fewer than four of them are base metal failures...
  44. D

    MHB Binomial Expansion Approximation for $\frac{1}{\sqrt{1 - A^2u^2}}$

    Use the binomial expansion to give the approximation $\frac{1}{\sqrt{1 - A^2u^2}}\approx 1 + \frac{1}{2}A^2u^2$ How can this be done? Using the definition for (x - y), we have $$ (x - y)^n = \sum_{k = 1}^{n}(-1)^k\binom{n}{k}x^{n - k}y^{k} $$ but $n\notin\mathbb{Z}$.
  45. J

    Poisson vs Binomial distribution.

    Hello PF This might be a fairly simple question to most of you, but I was given this problem (don't worry, I already solved it just wondering about something) Suppose the probability of suffering a side effect of a certain flu vaccine is 0.005. If 1000 persons are inoculate, find the...
  46. R

    MHB Binomial Distribution in the Exponential Family of Distributions

    A pdf is of the exponential family if it can be written $ f(x|\theta)=h(x)c(\theta)exp(\sum_{i=1}^{k}{w_{i}(\theta)t_{i}(x))}$ with $\theta$ a finite parameter vector, $c(\theta)>0$, all functions are over the reals, and only $h(x)$ is possibly constant. I would like to show the binomial...
  47. B

    Binomial probability with TI-84 binomcdf function

    Homework Statement If a fair die is tossed 7 times what is the probability of at least two 4s? Homework Equations The Attempt at a Solution To solve this I used my TI-84's binomcdf function. I just want to see it I'm doing it correctly. p(2<=x<=7) = 1 - p(0<=x<=1) = 1 -...
  48. T

    Why is the Binomial Formula a Derivation?

    I mean the binomial formula is something of the form ##\left(a+b\right)^n## = ##\sum_{i=1}^{n}\dbinom{n}{k}a^{n}b^{n-k}## and then you have the linear map ##\psi : A \rightarrow A## which is a derivation when; ##\theta(xy) = y\theta(x) + x\theta(y)## for all x,y in A so the leibniz formula...
  49. S

    Binomial Distribution and Selection of Suitable Values

    For binomial distributions, how can you tell which central tendency value (mean, median, or mode) and which variability value (interquartile range, variance, standard deviation, etc.) are most appropriate for the data? Thanks for any reply.
  50. Daaavde

    Differences between binomial distribution and forced probability distribution

    Differences between binomial distribution and "forced" probability distribution Hi everyone. Yesterday I was thinking about probability and real life and about the fact that we always expect life's facts to behave according to probability theory. If we flip a coin and we get 6 times heads...
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