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. V

    Summation of Products of Binomial Coefficients

    Homework Statement Find and prove a formula for sum{ (m1 choose r)(m2 choose s)(m3 choose t) } where the sum is over all nonnegative integers r, s, ant t with fixed sum r + s + t = n. Homework Equations The Attempt at a Solution I first attempted to find the number of combinations of r...
  2. M

    Binomial Coefficients Identity

    Homework Statement Prove that for an integer n greater than or equal to 2, nC1 - 2nC2 + 3nC3 - + ... = 0. (nCm means n choose m) Also, 2x1 nC2 + 3x2 nC3 + 4x3 nC4 +... = n(n-1)2^(n-2) Homework Equations (1+t)^a = 1 + aC1(t) + aC2(t^2) + ... The Attempt at a Solution I don't know...
  3. M

    Finding success probability given a binomial probability?

    Is there a way to calculate, say, the probability of a dice landing on an 11, given that the binomial probability of getting exactly six elevens in 100 tosses is 24.6%?
  4. Z

    Casio fx-9860G - calculating binomial coefficients and binomial distribution

    How to calculate 1) binomial coefficients and 2) binomial distribution on a Casio fx-9860G calculator?
  5. Z

    Binomial coefficient summation proof

    Homework Statement Prove that \sum^{l}_{k=0} n \choose k m \choose l-k = n+m \choose l Hint: Apply the binomial theorem to (1+x)n(1+x)m Homework Equations The Attempt at a Solution I apply the hint to that thing to get \sum^{n}_{j=0} n \choose j x^j \sum^{m}_{k=0} m \choose k...
  6. E

    Solving the Trinomial & Binomial Distributions: A Challenge

    can anyone help me please can anyone solve this problem for me please Q) The Binomial distribution allows the calculation of the probability of k successes in n trails where there are only two outcomes: success or fail with probabilities p and q respectively. The Binomial probability is...
  7. P

    Why Does the Binomial Theorem Summation Not Equal 2n When n Varies?

    Using summation((\stackrel{n}{k})xkyn-k) = (x+y)n, I let x = y = 1. This should then result in summation((\stackrel{n}{k})*1*1) = (1 + 1)n = 2n. Expanding the summation, I get (\stackrel{n}{0}) + (\stackrel{n}{1}) + ... +(\stackrel{n}{n}) = 2n. Solving this results in...
  8. N

    Normal Approximation to Binomial Distribution

    On a production line, only 45% of items produced meet quality standards. A random sample of 500 items will be taken. Using the normal approximation to the binomial distribution, approximate the probability that less than half of the sampled items meet quality standards. 500*.5 = 250...
  9. T

    Binomial Probability problem.

    Homework Statement 10% of engines manufactured on an assembly line are defective. If engines are randomly selected one at a time and tested, what is the probability that the first defective engine will be found between the 5th trial and the 25th trial, inclusive? Homework Equations...
  10. G

    Prove 2^n possibly with the binomial theorem

    Prove for all n\inN 2n= (\stackrel{n}{0})+(\stackrel{n}{1})+...+(\stackrel{n}{n}) So I used mathematical induction base case: n=0 so 20=1 and (\stackrel{0}{0})=1 induction step: Let n\inN be given, assume as induction hypothesis that 2n=...
  11. silvermane

    Solve Binomial Thm Proof: Prove Increasing & Bounded Sum

    Homework Statement Prove that (1 + 1/n)^n = 1 + \sum1/m!(1 - 1/n)(1-2/n)...(1-(m-1)/n) where our sum is from m=1 to n. 2. Attempt: I recognize the binomial theorem here, but I'm having a mental block in how to figure this out. I should know how to do this, I think I just need a little help...
  12. R

    Binomial theorem - not an easy question

    Hi guys, I'm Filip and as a 11th grade student I have a question about one mathematical problem. It says: If the coefficient of xk in the expansion of (3+2x-x2 )*(1+x)34 is zero. Find the value of k. I know it's something related with binomial theorem, but I don't really know how to start. Thank...
  13. M

    Practical use of binomial and Poisson Distribution in the field of engineering

    Hi... Hope i 'll get the good result that where we practically use the binomial and poisson distribution in the field of engineering...
  14. M

    Tidal Potential & Binomial Approximation

    Homework Statement There is a derivation in the text that I'm having problems replicating. The text gives the formula for tidal potential as: U_{tid}=-GM_{m}m(\frac{1}{d}-\frac{x}{d^{2}_{0}}) Where M_{m} is the mass of the moon, d is the distance from the CM of the moon to the point of...
  15. M

    Exploring Binomial Expansion in Electric Dipole Fields

    I'm learning the subject of electric fields from Resnick and Halliday's book, and they have an equation for the field of the dipole: E = \frac{1}{4\pi\epsilon_0}\frac{p}{x^3} \left[1+\left(\frac{d}{2x}\right)^2\right]^{-3/2} Their next step is to find out what happens when x is larger than...
  16. L

    Inductive proof of a binomial series

    Homework Statement Use mathematical induction and Pascal's Identity to prove: \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - ... + (-1)^{k}\binom{n}{k} = (-1)^{k}\binom{n-1}{k} The Attempt at a Solution First, I guess this means something like: \sum_{i=0}^{k}(-1)^{i}\binom{n}{i} =...
  17. P

    Binomial Probability: More Than 1 Survival from 10 Chicks

    Homework Statement For a certain species of bird, there is a chance of three in five that a fledgling will survive. From a brood of ten chicks, find the chance that more than one will survive. Let p = survival chance = 3/5 Let q = non-survival chance = 2/5 P(less than one will not...
  18. S

    Conditional Binomial Distribution

    How do I find a conditional bionomial distribution? For example, if I want the probability that k=7 (for instance, 7 could be any number depending on the experiment), given that k is greater/equal to 4. I know what the equation would look like i.e.: F(k=7|k >= 4)= P(k=7, k>=4)/P(k>=4)...
  19. D

    How to maximize P(Y = y*) for a negative binomial distribution

    How can I find probability p that maximized P(Y = y*) when Y has a negative binomial distribution with parameters r (known) and p? I've just reduced the problem with some algebra, but other than guess-and-check I have no rigorous way to solve this problem.
  20. Q

    Binomial alternating series- even numbers

    Homework Statement Ok, I know that (-1)^r \binom {n} {r} is supposed to equal 0. But I have plugged some numbers into this series, and this doesn't seem to be true for even numbers of n? Like for n = 4 and r = 4, I have: 1 - \frac{4!}{1!3!} + \frac{4!}{2!2!} - \frac{4!}{3!1!} +...
  21. O

    Prime divides its binomial coefficient?

    Hi all, this homework problem's been driving me nuts. It seems like it's probably pretty straightforward and I'm missing something obvious, but I just can't work it out. Homework Statement prove that if p is a prime number that p|B(p,m) where B(p,m) is the ordinary binomial coefficient...
  22. A

    Solving the alternating sum of binomial coefficients using telescoping series

    I havn't done this in a long time! And apparently I should know this easy, it sort of looks like a proof by induction, which I havn't done before and I am frantically trying to learn! Show that for each integer n the alternating sum of binomial coefficients: 1 - (n) + ... + (-1)^k(n) + ...
  23. M

    Probability using binomial distribution

    Homework Statement In a comm. system a byte (8 bits) is transmitted with a bit error probability of 0.1. If the system can correct at most one error made in each byte. a) what is probability of a byte being received correctly (after correction)? b)What is most probable number of errors...
  24. L

    Binomial Theorem Proof: (nC0)(mC0) + (nC1)(mC1) + ... + (nCm)(mCm) = (n+m C m)

    Homework Statement To Prove: (nC0)(mC0) + (nC1)(mC1) + ... + (nCm)(mCm) = (n+m C m) where nC0 = n choose 0 and so on. Homework Equations The Attempt at a Solution Tried expanding the whole thing using factorials - but didn't work. Any hints would be really welcome!
  25. D

    Binomial Distrubition/Die rolling Question.

    How to Find the Probability of rolling at least 2 "sixes" in 6 rolls of a balanced die. I am trying to solve using the Binominal Formula, P(X = r) = nCr p r (1-p)n-r But am not really sure what the probability rates for success and failure should be or how to compute it. Any advice? Thanks.
  26. C

    Caculating integral with binomial coefficient

    Homework Statement Homework Equations What is the integral of \int^{0}_{1} nCy x^{y} (1-x)^{n-y} dx ? The Attempt at a Solution \left(nCy\right) \int^{0}_{1} x^{y} (1-x)^{n-y} dx
  27. L

    Deviance of Binomial generalized linear model

    The formula for the deviance of a binomial generalized linear model is: D = 2\sum[y_i \log(\frac{y_i}{\hat{y}_i})+(n_i-y_i)\log(\frac{n_i-y_i}{n_i-\hat{y}_i})]. where the responses y are Binomial(n_i, p_i), and \hat{y}_i = n_i\hat{p}_i. The second log in that equation is undefined when...
  28. C

    Expanding (x+y)^0.5: Can the Binomial Theorem be applied to this generalization?

    How would you expand (x+y)^0.5 ?
  29. O

    Is there a mistake in my non-commutative binomial expansion for (A+B)^3?

    Homework Statement Well, I was trying to expand say for the third power of (A+B), where A and B are non-commutative. The Attempt at a Solution I get (A+B)^3=(A^2+AB+BA+B^2)(A+B)=A^3+ABA+BA^2+B^2A+A^2B+AB^2+BAB+B^3 but from a few sources online, it should be...
  30. Y

    Proving Binomial Distribution Expected Value & Variance

    hello, i need to prive that for a binomial r.v X E[X]=NP and VAR(X)=NP(1-P). I tried to prove it using the deffinition of expectation: E[x]=\sum xi \stackrel{N}{i} p^{i}(1-p)^{n-i} now what? thanks...
  31. T

    Probability question - Binomial distribution

    Homework Statement A game is played by tossing two unbiased coins repeatedly until two heads are obtained in the same throw. The random variable X denotes the number of throws required. Find the expression for P(X=r). Homework Equations The Attempt at a Solution It looks to be a...
  32. J

    Is it possible to calculate a binomial distribution with a non-constant p?

    Here's the actual problem I'm faced with. Suppose a segment of dna with 100 mutations (SNPs) which occur at different frequencies from each other and between 2 different populations for the same mutation. The expected number of mutations occurring in the segment of dna is different in either...
  33. T

    "Solving Coefficient of x^n in Binomial Expansion

    Homework Statement Find, in the simplest form, the coefficient of x^n in the binomial expansion of (1-x)^(-6). Homework Equations The Attempt at a Solution i am not sure how to go about with this.
  34. G

    Binomial series vs Binomial theorem, scratching my head for three days on this

    In my book, it says that the Binomial Series is \sum_{n=0}^{\infty }\binom{n}{r} x^n Where \binom{n}{r} = \frac{n(n-1)...(n-r+1)}{n!} for r\geq1 and \binom{n}{0} = 1 Now here is where it got to be, I know that the \binom{n}{r} = \frac{n(n-1)...(n-r+1)}{n!} were derived through the...
  35. Z

    A sum involving binomial coefficient

    sum_{i=k}^{n} {i \choose k}i^{-t} where t is a constant. Does it have a closed form?
  36. T

    Understanding the Binomial Expansion and its Relationship to e^p

    How is 1+p+\frac{p^2}{2!}+\frac{p^3}{3!}+...=e^p ?
  37. K

    Need help on simple binomial problem

    Hello everyone, Just have a quick question on a binomial problem. The problem is as follows: A teacher is giving a 15 question true-false quiz. He wants to design the quiz such that a person that guesses on all the answers have less than a 0.10 probability of passing. What should the...
  38. P

    Binomial sequence and graph display

    Homework Statement Given a series of 0 and 1 , how can we plot the binomial curve ?? Example: 00000011100010010100011110 say,p=0.8 q=0.2, N=26 Homework Equations If I apply the classic binomial formula, 26C0 (0.8)^0(0.2)^(26-0) etc.. seems cannot do so.
  39. G

    Expanding x^n-a^n without Binomial Theroem ?

    Homework Statement This is the given Theorem in my book, everything seems fine except that I cannot figure how they expanded (xn - an) Homework Equations The Binomial Theorem The Attempt at a Solution According to me (xn - an) = {[(x+a)-a]n - an} and expanding it would yield...
  40. B

    A little help with a binomial theorem proof

    Homework Statement (here, (n,k) reads n choose k) prove that (n,0) - (n, 1) + ... + (-1)n(n,n) = 0 Homework Equations binomial theorem The Attempt at a Solution so this proof is relatively straightforward when n is odd. it's just matching up terms and having them cancel each other...
  41. S

    Bounds for the mean of the minimum of binomial random variables

    Dear Friends, I want to find an upper and lower bound for the expected value of the minimum of independent binomial random variables. What paper/book do you suggest for this problem? In other words, I need to find bounds for E(min(X1,X2,...,Xn)), where Xi 's are independent random variables...
  42. Saladsamurai

    Deriving the binomial distribution formula

    I am trying to follow along with this derivation of the binomial distribution formula: b(x;n,p) = nCx*pxqn-x But I do not really understand the meaning of the part on bold. What is this "specified order" business now? I feel like I am missing something big here.
  43. S

    Was just looking at binomial theorem, i am confused [help]

    i was told the binomial theorem is as follows: (1-x)^n = 1-nx+ (n(n-1)/2!)x^2 - (n(n-2)/2!)x^3 ... not sure if this is right could some one clear this doubt for me any help is appreciated was told this in a physics class
  44. Jake1802

    Summation with Binomial Expansion

    Homework Statement How can i prove this relationship \sum _{i=0}^k \text{Binomial}[n+1,k-2i] - \sum _{i=0}^k \text{Binomial}[n,k-2i]=\sum _{i=0}^k \text{Binomial}[n,k-1-2i] Homework Equations Binomial (n,k)=n^k/k! The Attempt at a Solution I attempted subbing into mathyematica but this didn't...
  45. M

    By using binomial distribution if two coin are tossed 4 times ,find?

    by using binomial distribution if two coin are tossed 4 times ,find? 1)the probability of 2 heads in 4 times ? 2)the probability of less than one head once? 3)the probability of than 2 tails in 3 times ? 4)the expected number of two tails ? 5)the variance of the number of 2 heads?
  46. S

    Binomial expansion of relativistic formula

    Homework Statement The speed v of electrons from a high energy accelerator is very near the speed of light c. Given the voltage V of the accelerator, calculate the ratio v/c. The relativistic formulafor this calculation is (see relevant equations) Use the binomial series to find (1-v/c) in...
  47. A

    Binomial Expansion Question - fractional powers

    Homework Statement My question is simple is there a formula for the bi/tri-nomial expansion of bi/tri-nomials raised to fractional powers. that is, (x^{2}+1)^{1/2} or (x^{2}+x+1)^{1/2} I know pascals triangle for integer exponents but i can't really find anything about fraction...
  48. S

    Help proving with the Binomial Theorem

    Homework Statement (n¦0)-(n¦1)+(n¦2)-. . . ± (n¦n)=0 that reads n choose zero and so on Homework Equations Prove this using the binomial theorem The Attempt at a Solution I really have no idea where to start. Any help would be greatly appreciated thanks
  49. H

    Understanding Binomial Coefficients: Solving a Sample Problem

    I understand permutations, combinations and such, but I can't seem to make sense of binomial coefficients, or at least the notation. As an example, could someone walk me through the notation for a generic problem.. something like 100 people eligible for an award and the winner can choose 1...
  50. F

    Counting Combinations with Restricted Summation using Generating Functions

    If we have numbers 1,2,3,4,5,6,7,8,9,10,11. We want to pick 5 numbers out of that, but there is a restriction - the summation of the 5 picked numbers must be 21 or less. How many different combinations can we get? The answer is 24 but I would like to know how to work it out (besides...
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