Binomial distribution Definition and 145 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. A

    Engineering Statistics: Binomial Distribution

    Homework Statement Let k >= 3 be any integer. What is the probability that a random k-digit number will have at least one 0, one 1 and one 2? (as usual every number starts with either 1,2,...9 and NOT zero) Homework Equations b(x : n,p) = (n x)p^x*(1-p)^(n-x) where x = 0, 1, 2, ... ,n...
  2. Q

    What is the Probability of Fibrillation After N Attempts?

    Homework Statement The question provides a table and asks: Number of Attempts Fraction persisting in fibrillation 0 1.00 1 0.37 2...
  3. 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?
  4. 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...
  5. 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)...
  6. 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.
  7. 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...
  8. 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...
  9. 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...
  10. 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.
  11. 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?
  12. A

    Relationship b/w Binomial, CLT & Poisson Distrib.

    From the central limit theorem the binomial distribution can be approximated by a normal distribution N(0,1). But the binomial distribution can also be approximated by a poisson distribition. Does this mean there is a relationship between the normal distribution and the poisson distribution...
  13. A

    Probability Theory ; Binomial Distribution?

    Homework Statement Now you and your fiend play a different game. You flip your coin until it comes up heads the first time. Let X denote the number of flips needed. Your friend rolls its die until it comes up "3" or "5". The first try let Y denote the number of rolls needed. Assume X and Y are...
  14. S

    Probability generating function (binomial distribution)

    Homework Statement The probabilty generating funtion G is definied for random varibles whos range are \subset {0,1,2,3,...}. If Y is such a random variable we will call it a counting random varible. Its probabiltiy generating function is G(s) = E(s^{y}) for those s's such that E(|s|^{y})) <...
  15. A

    Binomial Distribution Probability

    Let X be a Binomial B(\frac{1}{2},n), where n=2m. Let a(m,k) = \frac{4^m}{(\stackrel{2m}{m})}P(X = m + k). Show that lim_{m->\infty}(a(m,k))^2 = e^{-k^2}. So far, I've found that P(X = m+k) = (\stackrel{2m}{m+k}) \frac{1}{4^m} Then, a(m,k)=\frac{m!m!}{(m+k)!(m-k)!}. But I have no...
  16. M

    Binomial distribution formulae?

    [b]1. Let the p.m.f. pf M be defined by f(m)=x/8, x=1,3,4. What is the mean of M? [b]2. n!/n-r*p^n*(1-p)^n-r [b]3. 3!*1/3^3*2/3^2=.59 This is not the correct answer!
  17. P

    Uncertainty of binomial distribution?

    Hi, I'd like to know how to find the uncertainty of a function that has two binomial distribution s, something like Signal = N(yes) - N(no) Set p = 0.6 for yes. My problem is that I do not know how to find the uncertainty for N(yes) and N(no), and do not know how to find the uncertainty...
  18. C

    Binomial Distribution: Solving for P(X=2, N=4), P(X=1), and P(N=4|X=1)

    Homework Statement Suppose that the conditional distribution of X given that N = n is binomial (n, 1/2) and the distribution of N is uniform over {2,4,6} a) Determine P(X=2, N = 4) b) Determine P(X=1) c) Determine P(N = 4| X =1) Homework Equations The Attempt at a Solution...
  19. G

    Prob. for average value or less in binomial distribution?

    Hello! Is there a closed form expression or a good estimate for the probability that a binomial distribution yield the average np or less. Basically I'm asking for a good way to evaluate P=\sum_{k=0}^{np} \begin{pmatrix} n\\ k \end{pmatrix} p^k(1-p)^{n-k} I just figured that for the...
  20. L

    Poisson distribution and binomial distribution questions

    Please help with this thanks :) 1. (a) Define the Poisson probability distribution with mean μ. (b) Write down the binomial distribution for x successes in n independent trials each with probability p of success. (c) On average, 0.15% of the nails manufactured at a factory are known to...
  21. B

    Calculating Binomial Distribution: Probability of Third Strike on Fifth Well

    Hi Guys, I have been given the probability that a drill strikes oil in a region = 0.2. I know that if I wanted to find the probabilty of say striking oil 3 times out of 5 wells It would be 5Choose3 = 5!/((2!)(3!)) * (1/5)3* (4/5)2 = 0.0512 My question is how would I go about...
  22. K

    Binomial distribution of children

    I'm trying to figure out this problem but i keep getting stuck. Homework Statement A woman wants to have a 95% chance for a least a one boy and at least one girl. What is the minimum number of children that she should plan to have? Assume that the event that a child is a girl and a boy is...
  23. M

    Binomial Distribution Probability Problem

    Homework Statement The mailing list of an agency that markets scuba-diving trips to the Florida Keys contains 70% males and 30% females. The agency calls 30 people chosen at random from its list. What is the probability that the first woman is reached on the fourth call? (That is, the first 4...
  24. 6

    How to transform this word problem into a binomial distribution equation

    Homework Statement Because not all airline passengers show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is 0.10, and the passengers behave independently a) What is the probability...
  25. 2

    Expected value and negative binomial distribution

    Hello, Can someone please tell me whether I can use negative binomial distribution for this question. "If there are 3 types of books in a bookstore and each book has an equal probability of being bought. What is the expected number of purchases to get all 3 books?" Using negative...
  26. P

    Binomial Distribution: Expected Gain for Flipping a Coin Four Times

    Homework Statement A coin can be flipped a maximum of four times The following conditions exist: H(first) = $1 H(second) = $2 H(third) = $3 H(fourth) = $4 Where H = Heads And first, second, third and fourth, refer to what order one head is obtained. What is the expected gain...
  27. B

    Binomial Distribution Statistics Problem

    Homework Statement Estimate the probability that, in a group of five people, at least two of them have the same zodiacal sign. (There are 12 zodiacal signs; assume that each sign is equally likely for any person.) Homework Equations P(X=k) = nCk * p^{k} * (1-p)^k{} The Attempt at a...
  28. R

    Confusion on Poisson and Binomial Distribution

    Hey guys, Can anyone please explain the differences between binomial and poisson distribution. THANK U>>>>>>>>>>>>>>>>>>>>>
  29. S

    Calculating Chances of Success with Binomial Distribution: A Homework Example

    Homework Statement I am trying to figure out if I have a 20% chance to get what I want (let it = x) and I have 6 chances to do so (n=6), I am curious how I set this question up to find out my chances of getting 'x' once out of the 6 times I try. Homework Equations Binomial Distribution...
  30. T

    Binomial distribution and probability problem

    Homework Statement A population has an average of 12 defects per 100 feet of wire sampled and inspected. What is the probability of finding 20 or fewer defects in a sample? Homework Equations I think I am supposed to use the binomial distribution b(x;n,p) The Attempt at a...
  31. F

    Probability of X in Binomial Distribution B(20, 1/52)

    http://img262.imageshack.us/img262/4669/rangevu0.jpg Ok so I think its asking for a binomial distribution of B(20, 1/52) Would this be the probability function of X? Also the range space of X is asked in a lot of these questions, but I've no idea on how to calculate it. I thought it...
  32. R

    Binomial distribution notation

    If the probability of a successful outcome is p and failure is q and there are n trials P(X=x)= ^n C_x p^xq^{n-x} and you write that as X~Bin(n,p) <--- How would I read that? (like what does the ~ mean?)
  33. G

    Expectation of Negative Binomial Distribution

    I am re-writing up some lecture notes and one of the proofs that E[X] for the negative binomial is r/p where r is the number of trials...The problem is there are a number of books that say r(1-p)/p is the correct expectation whilst others agree with 1/p Which one is correct...for what its...
  34. L

    Unbised estimator of Binomial Distribution

    [SOLVED] unbised estimator of Binomial Distribution I have no idea how to find such an estimator SupposeX_1, ..., X_n \sim Bern(p) find an unbiased estimator of p^m, for m < n Induction on m was a nasty mess that should not be expected. The power of m causes some problem when I try to go...
  35. Simfish

    How is the negative binomial the inverse of the binomial distribution?

    Can anyone give a user-friendly explanation? http://en.wikipedia.org/wiki/Negative_binomial_distribution#Properties We see that the binomial distribution measures the probability of X successes after n trials, whereas the negative binomial measures the probability of the trial number after...
  36. C

    What is the Probability of Gerry Infield Getting Two Hits in His Next 5 At Bats?

    Homework Statement Gerry Infield has batting average of 0.326 what is the probability that he will have two hits in his next 5 times at bat?Give your answer correct to 3 decimal places. The Attempt at a Solution It is a binomial distribution with n trails and x sucesses (hence, n-x...
  37. C

    Caculate the probability using a binomial distribution

    Ok so I have a problem I am not sure of the method I should use. In a recent survey, 60% of the population disagreed with a given statement, 20% agreed and 20% were unsure. Find the probability of having at least 5 person who agree in a mini-survey with 10 people. I tried to caculate the...
  38. S

    What is binomial distribution and how does it work?

    Hi, Can anyone explain binomial distribution to me. I tried wikipedia and some googling, but I just do not understand much of it. I don't come from maths background, I am more like an IT person. I need to write a short program calculating binomial distributions, however, first I need to...
  39. 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...
  40. 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.
  41. 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...
  42. 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...
  43. 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
  44. B

    Understanding the Bias in Binomial Distribution for Probability Calculations

    binomial distribution Prob of rolling a 1 = 1/10, rolling a 2 = 2/10, 3 = 3/10, 4 = 4/10 Let X be the value thrown Calculate E(X) and Var(X) To do this can't use E(X) = np and can't use Var(X) = npq is this correct?
  45. H

    Binomial Distribution (Statistics)

    Hi guys, if you can help me with this problem it would be of great help 1) Pythag-Air-US Airlines has determined that 5% of its customers do not show up for their flights. If a passenger is bumped off a flight because of overbooking, the airline pays the customer $200. What is the expected...
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