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. 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...
  2. L

    I Problems that could occur in estimating n from a Binomial distribution

    Hi, I am doing the following question: https://i.gyazo.com/f2e651334bcbd5f1dcb6d661e4160956.png I have estimated both n and theta. But the part that is throwing me off is what problem could you encounter in estimating n here? My only idea is that it might be something to do with the sample...
  3. J

    A Newton's Generalized Binomial Theorem

    I'm trying to expand the following using Newton's Generalized Binomial Theorem. $$[f_1(x)+f_2(x)]^\delta = (f_1(x))^\delta + \delta (f_1(x))^{\delta-1}f_2(x) + \frac{\delta(\delta-1)}{2!}(f_1(x))^{\delta-2}(f_2(x))^2 + ...$$ where $$0<\delta<<1$$ But the condition for this formula is that...
  4. Pushoam

    Why Does Calculating Binomial Probabilities Differ from Simple Outcome Ratios?

    Homework Statement I am not getting the above. Let me calculate the probability of getting 2 successes from 5 Bernoulli trials. There are total 10 possible outcomes as each trial has two possible outcomes. The probability of getting one success is P(S1) = No. of successes / no. of total...
  5. bdolle

    What is the Correct Way to Write a Binomial Expansion for (1+(1/x))^(-3/2)?

    <Moderator's note: Moved from a technical forum and therefore no template.> Hey All, For my modern physics class we were told to write out a binomial expansion of (1+(1/x))^(-3/2). I am fairly confident in the work I did, but my professor posted his work and it is different and way simpler...
  6. T

    Calculating Binomial Probability for Car Color Preferences

    Homework Statement Car colour preferences changes over year . In this year , suppose that 10% of the car are randomly selected , let the sample of cars are 20 . Find the probabilities between thre and five cars ( inclusive ) are black ... I am aksed to do this question using binomial ...
  7. D

    Fractional equations w/ binomial denominators

    Homework Statement D+8/D-2 = 9/4 See image, original equation in black. Homework EquationsThe Attempt at a Solution See image. Having a little trouble with this. Ive attempted to solve it two ways. The first was to multiply both sides by ##d-2## which gave me the correct answer of...
  8. lfdahl

    MHB Binomial coefficient challenge

    Prove the following identity:\[\sum_{n =1}^{\infty }\frac{1}{\binom{n+r}{r+1}}=\frac{r+1}{r},\: \: \: \: r,n \in \mathbb{N}.\]
  9. O

    Why isn't this a binomial distribution?

    Homework Statement An ordinary die is painted red on two sides, white on two sides and blue on two sides. Find the probability we get no reds in 12 rolls of the die. Homework EquationsThe Attempt at a Solution GENERAL QUESTION:[/B] I thought this would be a binomial distribution, but the book...
  10. K

    I How Does Probability Affect the Total Number of Coupons a Customer Can Expect?

    Hi all, I am solving a practical math problem. There is a sale in one of the shopping mall in my town. The mall gives 10 coupons to a new customer. The customer could use one coupon at a time and when it is used, one could spin a fortune wheel to win more 10 more coupons. If one doesn't win...
  11. lfdahl

    MHB Prove an identity with binomial coefficients

    Prove, that $\sum_{j=1}^{2n-1}\frac{(-1)^{j-1}j}{{2n \choose j }} = \frac{n}{n+1}$ i have tried with proof by induction, but it is very difficult to use this technique. I should be very glad to see any approach, that can crack this nut.
  12. I

    I Need closed form for a Binomial series

    Hello I was solving a problem in probability. Here is the statement. Seven terminals in an on-line system are attached to a communications line to the central computer. Exactly four of these terminals are ready to transmit a message. Assume that each terminal is equally likely to be in the ready...
  13. Schaus

    Binomial Theorem - Determine n

    Homework Statement The sixth term of the expansion of (x-1/5)n is -1287/(3125)x8. Determine n. Homework Equations tk+1=nCkan-kbk The Attempt at a Solution tk+1=nCkan-kbk t5+1=nC5(x)n-5(-1/5)5 This is where I'm stuck. Do I sub in -1287/(3125)x8 to = t6? If so what do I do from here...
  14. Schaus

    Solve Binomial Theorem Homework: Find Coefficients of Degree 17 & x7

    Homework Statement 1. Given the binomial (x2-x)13determine the coefficient of the term of degree 17. Answer = -715 2. Given the binomial (2x+3)10 determine the coefficient of the term containing x7. Answer = 414720 2. Homework Equations tk+1=nCkan-kbk The Attempt at a Solution #1 - What...
  15. C

    Help with basic binomial coefficient

    < Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown > Hello. I'm currently working my way through Lang's Basic Mathematics and cannot make sense of this question: Show that if n is a positive integer at most equal to m, then {m \choose n}+{m\choose...
  16. R

    B Issue with Binomial Expansion Formula

    Working through Leonard Susskind's book The Theoretical Minimum, I noticed an issue with his expansion for the Binomial Expansion (he was missing factorials in the denominators). This led me to some confusion about the final term that is generally written (bn). (a+b)n = an + nan-1b + n(n-1)/2...
  17. RJLiberator

    Binomial Distribution Question

    Homework Statement A good hitter in baseball has a batting average of .300 which means that the hitter will be successful three times out of 10 tries on average. Assume that the batter has four times at bat per game.a) What is the probability that he will get two hits or less in a three game...
  18. M

    MHB Binomial Expansion - Fractional Powers

    Hello! We know from 'Binomial Expansion' that (1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+... for \left| x \right|<1 . Why doesn't it work for other values of x? I can't understand this condition. I would be really grateful for clear explanation!
  19. Alettix

    B Binomial Expansion with Negative/Rational Powers

    Hello! When studying binominal expansion: ## (a+b)^n = \sum_{k=0}^{n}{{n \choose k}a^{n-k}b^k} ## in high school, we proved this formula with combinatorics considering that "you can choose either a or b each time you multiply with a binom". Probably, this is not a real mathematical proof at...
  20. J

    MHB Calculate Upper Bound for $\displaystyle a_{n}$ in Binomial Limit Evaluation

    Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{1}{\binom{n}{k}}$ is I have tried like this way:: Let $\displaystyle a_{n} = \sum^{n}_{k=0}\frac{1}{\binom{n}{k}} =...
  21. A

    I Summation for extended binomial coefficients

    Is there a way of writing summation(s) to obtain the extended binomial coefficients? i.e., Considering the expansion of (1+x+x^2+x^3+...+x^N)^M can we write expressions (presumably involving summation and/or product notation) for the coefficients (on x^j in the expansion of the above, for each...
  22. M

    MHB Comparing Binomial & Binary Heaps: What's the Difference?

    Hey! :o I am reading about binomial heaps. They are priority queue data structures similar to the binary heap, right? (Wondering) Which is the difference between these two heaps? (Wondering) I haven't really understood the operation of the binomial heaps Union. Could you explain it to me...
  23. L

    Radioactive decay, relation between binomial to expon. dist

    You can model the probability for radioactive decay as a Poisson distribution. This is the probability for radioactive decay within a specific time interval. (I probably got some of it wrong here). P(k,μ)=λ^k⋅exp(-μ)/k! Is there a way to use this formula to derive the other formula for...
  24. P

    MHB What Is the Probability That Any Bucket Receives Exactly n Red Balls?

    Greetings. I would need help as follow up to the answers to my http://mathhelpboards.com/basic-probability-statistics-23/prob-red-ball-buckets-binomial-17282-post79735.html#post79735. After having thrown R red balls over M buckets, the probability that exactly n red balls fall in one, specific...
  25. MOHD ZAKI

    I Approximation to sqrt(1+(d^2)/(x^2))

    using binomial theorem can I write sqrt(1+(d^2)/(x^2)) = 1+ .5(d^2)/(x^2)? d is a variable. X known constant.
  26. T

    I Damped Oscillators and Binomial theorem step

    I uploaded a picture of what I am stuck on. I understand the equation of motion 3.4.5a for a damped oscillator but I don't understand how to use binomial theorem to get the expanded equation 3.4.5b. I am no where near clever enough to figure this one out. I know how to use binomial theorem to...
  27. thegreengineer

    What Is the Probability That Exactly 3 Out of 8 Children Are Girls?

    Right now I'm having a problem with a statistics problem. More specifically with a binomial distribution problem. The problem says: There is a family composed by 8 children. Calculate the probability that 3 of them are girls As far as I know, binomial distribution formula says...
  28. P

    Binomial expansion for fractional power

    Homework Statement So, I'm solving a dipole thing and I have these vectors: |r + d - r'| = (r² + d² - r'²)(1/2) Homework Equations I want to expand this but I have no idea how! I know I may have an infinite power series, but I may expand at the square terms tops... Before I needed to do the...
  29. A

    MHB Binomial Experiments: Find Probability of x=5, x>=6, x<4

    Hi all, I'm a bit confused on this problem in my book. "Specify the values of n, p, and q and list the possible values of the random variable x. Sixty percent of U.S. adults trust national newspapers to present the news fairly and accurately. You randomly select nine U.S. adults. Find the...
  30. Amcote

    Stats: Approximating a binomial with a normal distribution

    Homework Statement A multiple choice test consists of a series of questions, each with four possible answers. How many questions are needed in order to be 99% confident that a student who guesses blindly at each question scores no more than 35% on the test? Homework Equations So I know that...
  31. amind

    Sum involving reciprocal of binomial coeffients

    Homework Statement If $$a_n = \sum_{r=0}^{n} \frac{1}{\binom{n}{r}}$$ Find $$\sum_{r=0}^{n} \frac{r}{\binom{n}{r}}$$ in terms of an and n 2. The attempt at a solution Let $$f(x) =\sum_{r=0}^{n} \frac{x^r}{\binom{n}{r}}$$ Then, an = f(1). Observe that f'(1) is the required sum. I was thinking...
  32. K

    MHB How Do You Calculate Binomial Probabilities for Survey Representation?

    Hello, I was hoping someone could help explain how to do this problem. I have been stuck on it for a while now. I know that you have to use a binomial with n=100, then n=1000 but I'm not sure how to set it up to solve for a range from 8-12%. Thanks! Any advice is appreciated. Also, for people...
  33. M

    MHB Solve Sum of {30 \choose i} with Binomial Theorem

    Simplify (find the sum) of {30 \choose 0} + \frac{1}{2}{30 \choose 1}+ \frac{1}{3}{30 \choose 2} + ... + \frac{1}{31}{30 \choose 30}. Do this is two ways: 1. Write \frac{1}{i+1}{30 \choose i} in a different way then add 2. Integrate the binomial thorem (don't forget the constant of integration)...
  34. C

    Calculating Variance of Y using Poisson and Binomial Distributions

    Homework Statement I need to find the variance of Y. The number of errors on a page follows a Poisson distribution with lambda = 0.40 average . Y = the number of pages without error among the first 112 pages . Homework EquationsThe Attempt at a Solution In Poisson, I know that Variance =...
  35. 3301

    Expand powers of binomial expressions

    Homework Statement The problem comes in second term after 4(2z...) Homework EquationsThe Attempt at a Solution so i got 4 (8z^3) 5k. 32 (z^3) * (20k). After that i left it like that but I supposed to get answer like this 160 z^3 * k I can conjoin constants of z and k variables? Or am I...
  36. M

    Binomial Expansion Practice Problems: Multiplying Binomials

    Homework Statement Homework Equations The Attempt at a Solution After using this formula I got 1-3x + 6x2 for (1-x)-3 and 1+x - 1/8 ∙ 4x2 for (1+2x)1/2 for the second part of the question I'd assume that you're supposed to multiply the equations? I think this done by timing the whole of...
  37. Q

    DiffEq, Binomial Expansion and limits

    Homework Statement Use algebra to show that U(x) = −√x − 1 and L(x) = −√x satisfy the ’funnel condition’ U(x) − L(x) → 0 as x → ∞ Homework Equations Funnel condition: The two fences come together asymptotically, i.e. U(x) − L(x) is small for large x. The Attempt at a Solution I think that...
  38. sinkersub

    Inverse Binomial Expansion within Laurent Series?

    Homework Statement Find the Laurent Series of f(z) = \frac{1}{z(z-2)^3} about the singularities z=0 and z=2 (separately). Verify z=0 is a pole of order 1, and z=2 is a pole of order 3. Find residue of f(z) at each pole. Homework Equations The solution starts by parentheses in the form (1 -...
  39. P

    MHB Prob of red ball in buckets. Binomial?

    I know for many of you this sounds trivial. For me it's not... A barrel contains M buckets each containing B balls, for a total of N = M x B balls. N is a big number. A small number R > 1 of balls are red. All the others N-R are white. The location of each one of these R balls is random...
  40. Orange-Juice

    Applying binomial theorem to prove combinatorics identity

    Homework Statement Prove that \sum\limits_{k=0}^l{n \choose k}{m \choose l-k} = {n+m \choose k}Homework Equations Binomial theorem The Attempt at a Solution [/B] We know that (1+x)^n(1+x)^m = (1+x)^{n+m} which, by the binomial theorem, is equivalent to: {\sum\limits_{k=0}^n{n \choose...
  41. A

    What is the closed form for the sum of binomial coefficients over any interval?

    Is there a way to find the following sum in closed form: ∑K(N,n) , where K(N,n) is the binomial coefficient and the sum can extend over any interval from n=0..N. I.e. not necessarily n=0 to N in which case on can just use the binomial theorem.
  42. alexmahone

    MHB Sum of binomial coefficients multiplied by k^2

    Evaluate \sum\limits_{k=1}^{12} {12\choose{k}}k^2 The answer is 159744.
  43. Y

    MHB Binomial distribution and conditional probability

    Hello all. I saw this problem in a book. I tried solving it, and compared it to the suggested solution. Results don't match, and I think that I am correct. Could you please help me decide what the right answer is ? This is the question: When coin 1 is flipped, it lands on heads with...
  44. SSGD

    Binomial Distribution for successive events

    So I new to probability and need someone to help me out if you could. I wanted to look into the probability of a process being complete if each operation of the process has its own likely hood of success or failure. I know that I should be using a binomial distribution to study the process...
  45. Destroxia

    Bernoulli Binomial Distribution

    Homework Statement Derive the bernoulli binomial distribution by generalizing the probability of a coin flip. ## P(k, n) = \binom{n}{k}p^{k}q^{(n-k)} ##, q = p - 1 Homework Equations Combination: ## \binom{n}{k} = \frac {n!} {k!(n-k)!} ## Prob. of coin flip: ## \frac {\binom{n}{k}} {2^n}...
  46. Odious Suspect

    Binomial Coefficient Factorial Derivation

    A few decades ago my algebra teacher showed how to construct the expression for binomial coefficients. If I start with Pascal's recursion, and propose C(n,k)=n!/k!(n-k)!, I can prove it to be so through induction. But that doesn't give me that happy feeling that comes with understanding. It...
  47. JonnyMaddox

    Exploring the Binomial Formulas & Beyond

    Hey JO, You all know the binomic formulas I guess. Let's look at the first: (a+b)^2=a^2+2ab+b^2 Now this can be interpretet as the area of a square with the sides (a+b). And that means the area of the square is decomposed into the components a^2,2ab and b^2. And this can also be done for a cube...
  48. G

    Use of binomial theorem in a sum of binomial coefficients?

    Homework Statement How to use binomial theorem for finding sums with binomial coefficients? Example: S={n\choose 1}-3{n\choose 3}+9{n\choose 5}-... How to represent this sum using \sum\limits notation (with binomial theorem)? Homework Equations (a+b)^n=\sum\limits_{k=0}^{n}{n\choose...
  49. T

    Binomial distribution of coin tosses

    Homework Statement 1. A fair coin is tossed 100 times. (a) Find an approximate probability of getting at least 60 heads. (b) Find an approximate probability of getting exactly 60 heads. The Attempt at a Solution part b) would be b(60;100,.5) part a) we would need the table for the cumulative...
  50. sunrah

    Binomial -> Poisson distribution question

    Homework Statement A teacher has an infinite flow of papers to mark. They appear in his office at random times, at an average rate of 10 a day. On average 10% of the manuscripts are free from errors. What is the probability that the teacher will see exactly one error-free manuscript (a) after...
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