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. B

    Approximating an expression with the binomial expansion

    f_r=(\frac{1+v}{1-v})f_i For an automobile moving at speed v that is a small fraction of the speed of light, assume that the fractional change in frequency of reflected radar is small. Under this assumption, use the first two terms of the bionomial expansion (1-x)^n\approx{1-nz \mbox{for}...
  2. C

    Applications of the Binomial Theorem

    How would you quickly derive the binomial series? Would you have to use Taylor's Theorem/ Taylor Series? And does the Binomial Theorem follow from the binomial series? Are there any applications at all of the binomial series/ Binomial Theorem to special relativity? I know the binomial series is...
  3. X

    Special Relativity and Binomial Expansion

    Hi all, I am working on the last part of a problem now in which I am trying to find what velocity (as a fraction of c) must be traveled from the Earth to Andromeda (a distance of 2.00x10^6 light-years) in order for only 20 years to pass in the reference frame of the rocket. I created my...
  4. E

    Is 2^n a Divisor of Binomial Coefficient B(2^n, m)?

    I am trying to prove (or find a counterexample for) this: Let n be any positive integer, and m any odd integer, with 1 <= m < 2^n. Also, let B(x,y) denote the binomial coefficient, x! /( y! ( x - y )! ). Then 2^n | B( 2^n , m ). Any help is welcome.
  5. M

    Solving the Binomial Theorem: (2x^2 - 1/x)^10

    1. For each of the following, simplify so that the variable term is raised is to a single power: (a) State the General Term tr in the Binomial expansion of (2x^2 - 1/x)^10 (b) Find the 7th term in the expansion (c) Is there an x^5 term? Find its coefficient. (d) Is there a constant term...
  6. A

    Solving Binomial Theorem Qs: If nC0 + nC1 + nC2...+ nCn = 256

    i an havign trouble solving this qs if nC0 + nC1 + nC2 +...+ nCn = 256 find the value of n all help appreciated:smile:
  7. E

    Binomial coefficient modulo a prime

    A question: Let bin(a,b) denote the binomial coefficient a! / ( b! (a - b)! ). Is it true that bin( 2p, p ) = 2 (mod p) if p is prime and p>=3 ?
  8. J

    Binomial and fibonacci heap type

    Suppose you have n elements with integer keys and they are to be put into a heap. What would be the time for creating a heap by repeated insertion into into an initially empty heap? Say, for instance if we are using binary, binomial and fibonacci heap type. Any suggestions?
  9. W

    Binomial distribution - killing cells with x-rays

    Dear Fellow mathematicians and Physicists,I am doing some MC modelling on tumour growth and radiotherapy treatment modelling and would like to know: Who out there would agree (or suggest alternatives) to the theroy that the chance of a cell being damaged/hit with radiation (and therefore...
  10. S

    How Does Binomial Expansion Work for Rational Indices?

    Hi I wanted to know what is the expansion of (1+x)^n when n is a rational number and |x|<1 ... Please let me know as soon as possible.. Thanks for your help Sincerely Sparsh
  11. M

    Derivation of the probability distribution function of a binomial distribution

    Is there a way to derive P (X=r) =^nC_r p^r q^{n-r} , r= 0, 1, 2,..., n where X: B(n,p) where n is the total number of bernoulli experiments, p the probability of success q, the probability of failure.
  12. N

    Probability and binomial distribution question

    There was a question on the test with the following information (binomial distribution) n=10 p=.2 Find the probability that X is : a. At least 3 b. At most 3 For part a I did P(X>=3)=1-P(X<=2) For part b I did P(X<=3) : \sum_{x=0}^3 perm(n, x)*p^x*(1-p)^(n-x) The last part is...
  13. R

    How Do You Calculate the Probability of Getting at Most One Brown M&M?

    Okay so I did this problem and got it wrong but I get one more chance to get it right. I tried using Binomial Dist to solve it but I failed. 30% of all M&Ms are brown. If 7 M&Ms are randomly selected, what is the probability that at most 1 is brown? I thought I would use 0 and 1 but I...
  14. N

    Is this correct please - Binomial Theorem

    I'm a complete beginner at these: (3n+1)^3 = (3n)^3 + 3(3n)^2 + 3(3n)^2 + 1 which would give me = 3 (9n^3 + 6n^2) +1 is this correct?
  15. N

    Binomial Theorem of (3n + 2)^x

    Could anyone tell me what (3n + 2)^x equals to please so I can check my answer? I get something awful that would take me too long to type
  16. P

    How Do You Find the Constant Term and Coefficient of x in a Binomial Expansion?

    I'm having problems figuring out how to do part (b) of this question. a) expand (1-2x)^3 and (1+1/x)^5 b) Find, in the expansion of (1-2x)^3 (1+1/x)^5 i) the constant term ii) the coeffecient of x I've done part a, and I know the formula for a general term for an expansion of a...
  17. S

    How Can Binomial Distribution Be Solved Without Using a Computer Program?

    Hi! Does someone know how to solve this equation (see the link) if all variables are known without P_U (without using a computer program)? http://www.itl.nist.gov/div898/handbook/prc/section2/gifs/pueq.gif Can it be done in some easy way? I have read courses in calculus at the...
  18. W

    Binomial Expansion: Coeff. of x^n in (1+x)^n/(1+2x)^2

    By writing (1+x) as \frac{1}{2}\left[1+\left( 1+2x\right) \right] or otherwise, show that the coefficient of x^n in the expansion of \frac{\left(1+x\right)^n}{\left(1+2x\right)^2} in ascending powers of x is \left(-1\right)^n\left(2n+1\right). -- I've tried expressing (1+x)^n as...
  19. -Job-

    Binomial Random Variable With Non-Integer value

    So the problem gives a binomial random variable X with parameters n=5 and p=0.25 and ask for the probability P(X=1.5). The binomial probability mass function is defined only for integers. Should i approximate using the normal distribution or the poisson?
  20. S

    Solve the Binomial Theorem: Find a, b, n

    anyone could help me with this question... in the expansion of (ax + by)^n, the coeffiients of the first 3 are 6561, 34992, and 81648., Find the value if a, b, and n. i did this... t1 = nC0 (ax)^n = 6561x^n a^n = 6561 t2 = nC1 (ax)^n-1 (by)^1 = 34992x^n-1y bna^n = 34992 but I'm not...
  21. B

    Solving a Complex Equation Using Binomial Formula with Cosine and Sine Functions

    Hello I'm suppose to show Given z^4 + z^3 + z^2 + z + 1 = 0 where z = cos(\frac{2 \pi}{5}) + i sin(\frac{2 \pi}{5}) by using the binomial product formula. r^n - s^n Is that then if r,s = z then z^4 - z^4 = 0 ? Sincerely and Best Regards Bob
  22. B

    How Accurate Is the Binomial Approximation for Small x Values?

    Show that if x is small then 1/(1+x) - root(1-2x) ~= (3/2)x^2 im not sure how to even begin this question. there was a part 1 but i don't think its relevant. Small numbers just confuse me...how small is small in any case?
  23. M

    Binomial Expansion: Calculating Constants \alpha and \beta

    Hello, another dull question on binomial expansion (approximation). I cannot follow the derivation for the approximate values of the two constants \alpha and \beta. (Text on propagation coefficient of TEM waves in transmission lines - constants of attenuation and phase-shift) Given \gamma...
  24. Z

    Solve Binomial Expansion: (2/x^2-x)^6 - No x Term

    I understand how Binomial expansion works, but I don't understand how to solve this problem. Give the term of (2/x^2-x)^6 that has no x.
  25. B

    Binomial Theorem - small values of x and approximate values

    "Show that for small values of x, the function (1+x)^(-1/2) may be approximated by 1-(1/2)x+(3/8)x^2 Hence obtain the approximate value of 1/root(1.01) to 4 decimals." im totally clueless. the example we have isn't well explained at all. can someone even just start me off...
  26. M

    Solve Binomial Identities to Approximate Fresnel Zone Radius

    I was going the derivations for Fresnel zone radius approximation, and there was a jump in the math which I don't fully understand. If someone could take a look at this and help me figure. I was hinted that it had to do with the binomial theorem, but I have no idea >.< Seems like LaTeX isn't...
  27. M

    Understanding Binomial Expansion: A Powerful Tool in Algebra and Calculus

    What is binomial expansion?
  28. M

    Simplest Explanation of Binomial Expansion

    could anyone explain binomial expansion in the simplest way?
  29. G

    Understanding Binomial Coefficients: Exploring the Formula and Its Applications

    k, maybe wrong forum... whatever... Anyway, so i was hoping someone could maybe derive or at least explain binomial coefficients. Like, i know that binomial(n,r)= n!/(n-r)!r! but why? in class the guy was explaining something like, if you're counting, and you're trying to arrange 3 balls into...
  30. L

    Binomial Theorem Application in Cauchy's and Sellmeier's Equations

    I am trying to do a question from Eugene Hecht's Optics book, which goes something like this: Given the following equations: Cauchy's Equation: n = C_1 + \frac{C_2}{\lambda^2} + \frac{C_3}{\lambda^4} + ... Sellmeier's Equation: n^2 = 1 + \sum_{j}...
  31. S

    Using the Binomial Theorem, find the first 5 terms in the expansion and estimate

    Okay, well, perhaps I am already done, perhaps not. This is why I seek your wisdom :) If you are asked to find the first 5 terms of the expansion (1 + 0.07)^9 using the Binomial Theorem, and then asked to use these terms to estimate the value of 1.07^9 what would you put? I added the 5...
  32. A

    Binomial coefficients and pascal's triangle

    I am working through a mathematics olympiad problem book, and I am asked to prove that n choose r, where n is the row number and r is the term number in the row is equal to that term. Can someone please give me a hint? I have not been able to find ANY proofs on the internet through a basic...
  33. M

    Discovering the Term Containing x^{20} in Binomial Theorem (2x - x^4)^{14}

    Find the term containing x^{20} in (2x - x^4)^{14}. I went t_{k+1}= _{14}C_{k}(2x)^{14-k}(-x^{4})^{k} = 2x^{14-k}(-x^{4k}) First of all, am I on the right track? If so what exactly do I do from there?
  34. S

    Exploring the Binomial Series: \alpha < n?

    hello all I thought this might be an interesting question to ask, consider the following series \sum_{n=0}^{\infty}\left(\begin{array}{cc}\alpha\\n \end{array}\right)x^{n}=(1+x)^{\alpha} this is known as the binomial series, what's confusing me is that how could this series exist when...
  35. K

    Binomial Distrinution - is my answer reasonable?

    Q: The simplest error detection scheme used in data communication is parity-checking. Messages sent consists of characters, each character consisting of a number of bits, 0 or 1. In parity checking, a 0 or a 1 is appended to the end to make the number of 1's even. The receiver checks the number...
  36. X

    . Binomial Theorem Question/Help .

    Hello Guys, I just have a few quick question on binomial theorem, any help would be greatly appreciated. 1. Expand using the binomial theorm in powers of x up to and including x^3 : (1 + 2x + 3x^2)^5 : I always thought binomial theorm would be used to expand binomials... this is not a...
  37. Cincinnatus

    Proving the Binomial Theorem with Induction

    It seems like this shouldn't be too difficult and yet I'm stumped. I am trying to prove the binomial theorem. (x+y)^n = the sum from k=0 to n of (x^k)*(y^n-k)*(The binomial coefficient n,k) Sorry, about the notation... Anyway, I figure the best way to go about proving this is by...
  38. O

    Calculating Probabilities for Multiple Colors of M&M's from a Bag

    I have a bag of M&M's that is 22.5% Blue, 12.5%Brown, and 65% other. If I pull 12 M&M's from the bag, what is the probabiliity that exactly 2 are blue and 3 are brown? I used the binomial to find the probability of 2 blue and 3 brown, and I want to multiply them together to get the answer...
  39. D

    How can the binomial theorem be proven?

    Hi! I haven't found any good proofs of the binomial theroem. But I've discovered how to go from (a+1)= bla bla to (a+b) = bla bla. So if anyone could told me how to prove (a+1) = bla bla...
  40. T

    Binomial Theorem: Coefficients and Expansions

    The questions are as follows 1) How many terms are there in the expansion of (a+3b-2c-d)^8 before like terms are combined? 4^8 2) How many terms are there after like terms are combined? _8C_4=70 3) What is the coefficent of a^2b^3cd^2? (_8C_2)( _6C_3)( _3C_1)( _2C_2)...
  41. L

    Finding the Independent Term in a Binomial Expansion

    Hi, I'm having some problems with this question - Find the term independent of x in (x^2 - 2/x)^6 I know the answer is something like 6Csomething 2^something, but I'm not sure how to get that. So far I've only really done simple things like (x+y)^n where y is an integer and not...
  42. K

    Solve Binomial Series: Quickly Answer Before Spring Break!

    Can Someone Plz Answer Asap! My teacher gave me homework and i got to convert for ex. (a+b)squared=a squared+ab+b squared, i got to do this up until to the 15th power. Please anyone have an ez way to do this or a shortcut. PLZ REPLY MY SPRING BREAK IS ALMOST OVER I HAVE LIKE 3 DAYS TO DO THIS
  43. V

    Poisson Approximation to Binomial

    For a binomial distribution with n=10 and p=0.5 ,we should not use the poisson approximation because both of the conditions n>=100 and np<=10 are not satisfied. SUppose we go way out on a limb and use the Poisson aproximation anyway. Are the resulting probabilities unacceptable...
  44. R

    Help Needed: Finding Voltage at x with Binomial Expansion

    I've been struggling with this problem for hours. I don't know what I did was wrong. Someone please give me a hint or point out where I did was wrong. Thanks a lot. There are three charges : the first charge of +2Q locates at (0,0), the second charge of -Q locates at (0,d), the third charge...
  45. C

    Which Model Better Values American Options: Binomial or Black Scholes?

    Is this a good problem: Which is a better model for valuing American options: the binomial or the Black Scholes? Any feedback is appreciated Thanks :smile: ps: sorry for the hiatus
  46. M

    Binomial Series Help: Expansion and Coefficients Explained | 1-x^-^3 Question

    I've got 1 question regarding the binomial series which I am currently stuck at. 1. Expand (1-x)^-^3 and express the coefficient of x^r in terms of r. What i did was to first expand it, according to the binomial series, and I got, 1+3x+6x^2...\frac {(-3)(-3-1)...(-3-r+1)}{r!}...
  47. F

    Find Coefficient of X^5 in (3x^3 - 1/x^2)^10

    Find the coefficient of the term X^5 of the expansion (3x^3 - \frac{1}{x^2})^{10} Another question off the topic. Find the x-coordinate of the minimium point of y=2x^2-5x+3 I know I have to complete the square but I'm not sure how its done.
  48. G

    Understanding Binomial Coefficients: Finding the r-th term formula

    I'm working on an IB mathematics portfolio, and here's a problem I don't understand: It's not specified what a and b is supposed to represent. That's where I need some explanation. I already made a nice Pascal triangle all the way to the 10th row. All help is appreciated.
  49. recon

    Expand & Simplify Binomial (1 + $\sqrt\frac{2}{n-1}$)^n

    How do you expand and simplify (1 + \sqrt\frac{2}{n-1})^n? I know this involves a binomial expansion and I can expand it to look something like \left(\begin{array}{c}n&0\end{array}\right){\frac{2}{n-1}}^\frac{0}{2} + \left(\begin{array}{c}n&1\end{array}\right){\frac{2}{n-1}}^\frac{1}{2} +...
  50. B

    Binomial distribution smallest value

    could someone please shed some light upon the following dilemma: Given that D~B(12,0.7), calculate the smallest value of d such that P(D>d) <0.90. much obliged
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