# What is Binomial distribution: Definition and 145 Discussions

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. ### I Write probability in terms of shape parameters of beta distribution

Assume that players A and B play a match where the probability that A will win each point is p, for B its 1-p and a player wins when he reach 11 points by a margin of >= 2The outcome of the match is specified by $$P(y|p, A_{wins})$$ If we know that A wins, his score is specified by B's score; he...
2. ### Understanding the meaning of "expected fraction" (Statistics)

The first part of the question asked me to calculate the mean and standard deviation for the number of remain votes in the simple binomial model consisting of total sample size of 2091 people. I believe this is fairly straightforward, it was simply ##E(X) = \mu = 2091(0.5) = 1045.5## votes and...
3. ### I Juicy game probability calculation

This Q was asked in a chat room: Find the number of pulls (draws from the distribution) required to have a 50% chance of getting a full set of event 5*s (*'s denote rarity in the game) from the Magic Tower Summon (of the Empires & Puzzles mobile game) which has 5x event 5*s each at a 0.2% drop...
4. ### MHB Binomial Distribution: Likelihood Ratio Test for Equality of Several Proportions

$\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: A survey of voter sentiment was conducted in four midcity political wards to compare the fraction of voters favoring candidate $A.$ Random samples of $200$ voters were polled in each of the...
5. ### I Binomial distribution of "worlds" in MWI

If we have a spin measurement with P(up)=0.5 en P(down)=0.5, this is equivalent to tossing a coin P(heads)=0.5 and P(tails)=0.5. The probability of having five heads and five tails out of ten tosses is the binomial: ##\binom{10}{5}(0.5)^5(0.5)^5##. So the same would hold for the spin...
6. ### B Poisson Distribution - Selecting cookies that are indistinguishable

Here's the problem: A chef made 500 cookies randomly mixed with 1000 nuts including 600 almonds and 400 hazelnuts in which each nut is the same size. Suppose the number of pieces of nuts in a piece of cookie follows a Poisson distribution. (a) Suppose cookies are randomly selected one-by-one...
7. ### B Coin Tossing: Binomial Distribution Explained

Why tossing a coin three times is said to have binomial distribution? I'm little bit confused.
8. ### B Does the binomial distribution play a role determining p from data?

In a game heroes have a maximum dodge rate, from experimental data we have 13 dodges out of 24 attacks (so 11 hits). A fellow on my discord server had immediately solved for the dodge rate as being 13/24. I started to explain it is not so simple as dividing (24-11)/24=13/24 is not the dodge...

10. ### B How to handle probabilities of the number of trials in a Binomial distribution

Suppose our process has a 85% chance of 2 trials and a 15% chance of 3 trials, and the rest is straightforward binomial distribution, do I take the weighted average of the binomial distribution at 2 and 3 trials? This is for a game so, yeah thanks.
11. ### I Generating samples on a 2-D composite binomial distribution

I would like to generate (X,Y) pairs such that they would follow a distribution something like this: This is the sum of three normal distributions. Each distribution could have a different taper along the X and the Y, plus an offset along X and/or Y. So the parameters of these three...
12. ### A Does this integration have a closed form solution?

I was trying to solve a differential equation that I defined to study the dynamics of a system. Meanwhile, I encounter integration. The integration is shown in the image below. I tried some solutions but I am failed to get a solution. In one solution, I took "x" common from the denominator terms...
13. ### Two teams, A and B, are playing a series of games

My attempt I used negative binomial to solve the problem, however I'm left with a polynomial that is difficult to solve? Is there any other way to approach this problem? I used the inequality because I'm trying to find the range of p. Since the probability of winning the series for team...
14. ### Variance of binomial distribution

Homework Statement Random variable Y has a binomial distribution with n trials and success probability X, where n is a given constant and X is a random variable with uniform (0,1) distribution. What is Var[Y]? Homework Equations E[Y] = np Var(Y) = np(1-p) for variance of a binomial...
15. ### Expected value of binomial distribution

Homework Statement A random variable Y has a binomial distribution with n trials and success probability X, where n is a given constant and X is a uniform(0,1) random variable. What is E[Y]? Homework Equations E[Y] = np The Attempt at a Solution The key is determining the probability of...
16. ### Am I justified in using the binomial distribution?

Homework Statement 12 non-distinguishable attacks from President Snow land in Panem’s 12 districts in a particular week. Assume the attacks are located randomly, with each configuration of attacks equally likely. What is the probability that some district had more than 1 attack? Homework...
17. ### 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...
18. 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...
19. ### 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...
20. ### 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...
21. ### 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...
22. ### Calculating Overbooking Probability for a Plane with Independent Passengers

Homework Statement A travel agency knows from experience, that 5% of passengers who have booked a ticket will not show up for the flight. Therefore the company sells 260 tickets for a plane that can only take 255 passengers. What is the probability that all of the passengers arriving for the...
23. ### Probability for a binomially distributed variable X

Homework Statement Let ##X \sim Bin(n, p)## where ##n=20## and ##p=0.1##. Calculate ##P(|X-\mu| \leq \sigma)##. Give your answer up to three decimal places. Homework Equations For a binomially distributed random variable, using moment generating functions we have: \mu= E(X) =...
24. ### I Why do we require conditions for the Poisson Distribution?

Three conditions must be met in order for the Poisson Distribution to be used: 1) The average count rate is constant over time 2) The counts occurring are independent 3) The probability of 2 or more counts occurring in the interval $n$ is zero Simply, why must these conditions be met for valid...
25. ### Binomial distribution problem

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...
26. ### Nuclear decay of small amount and by different processes

Hello I was reading this post, https://www.physicsforums.com/threads/nuclear-decay-of-a-small-number-of-atoms-calculation.853664/ and I wander if the binomial distribution is still a good model if you have a small amount of nuclei, and in addition they can decay by different processes (each...
27. ### 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...
28. ### 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...
29. ### 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}...
30. ### 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...
31. ### What is the probability of passing a shipment using a binomial distribution?

Homework Statement A company is interested in evaluating its current inspection procedure on large shipments of identical items. The procedure is to take a sample of 5 items and pass the shipment if no more than 1 item is found to be defective. It is known that items are defective at a 10%...
32. ### Chernoff Bound for Binomial Distribution

Hello, I've read in a paper that the following binomial distribution \sum_{k=floor(N/2)+1}^N{N\choose k}\varepsilon^k(1-\varepsilon)^{N-k} can be upper bounded using Chernoff bound by e^{ floor(N/2)}\,\Phi(s_0) where \Phi(s)=\left(1-\varepsilon(1-e^s)\right)^N and...
33. ### Lower Bound on Binomial Distribution

Hello all, Is there any lower bound on the following Binomial distribution \sum_{k=floor(N/2)+1}^N{N\choose k}\epsilon^k(1-\epsilon)^{N-k} as N goes to infinity and where epsilon is less that or equal 10^-3? Thanks
34. ### MHB Binomial Distribution Issue

A test consists of 10 multiple choice questions with five choices for each question. As an experiment, you GUESS on each and every answer without even reading the questions. What is the probability of getting exactly 6 questions correct on this test? The answer is: \binom{10}{6} (0.2)^6...
35. ### Question on Probability involving binomial distribution

Homework Statement P is the probability that a person aged x years will die in a year. Find the probability that out of 5 men A,B,C,D and E, each of x years, A will die in the year and be the first to die. Homework EquationsThe Attempt at a Solution I fixed A in the first place with...
36. ### S1 Probability Coin Toss

Having trouble with certain binomial and geometric distribution questions, which is indicating that my understanding isn't completely there yet. Any help would be greatly appreciated. 1. Homework Statement A bag contains two biased coins: coin A shows Heads with a probability of 0.6, and coin...
37. ### MHB Binomial distribution regarding: (≤, >, etc.)

Question is as follows: (a) = 0.4114 is the answer. Yet all I see from this answer is that X is simple equal to "0.4114". If it is "X ≤ 3" shouldn't "0.2061", "0.0692", and "0.0115" contribute to the answer somehow because they are "<" smaller than 3? I feel like I may be missing a...
38. ### Using approximations to the binomial distribution

Homework Statement This is the problem I am given. . It is in he picture below or in the thumbnail. I was also told that since ##n## is big enough that I can use normal approximations. Homework EquationsThe Attempt at a Solution I think that ##C_{\alpha}=C_{0.1}=2.33## which I got off the...
39. ### Binomial Distribution for a person walking in straight line

Homework Statement Can I measure the probability of a person being at a certain end location after n steps using the binomial distribution if, probability student goes x=x+3 is 0 <= p <0.5 , x=x-1 is 0<= 0.5 p <1.Homework Equations x=x+3 is 0 <= p <0.5 x=x-1 is 0<= 0.5 p <1 The Attempt at a...
40. ### Does this make sense? A binomial distribution with a twist.

I'm working on a project studying sea ice in the Arctic ocean. A brief overview of the essentials: The ice pack over the Arctic begins shrinking every summer beginning around June 1st, and begins to recover around Sep 15th. I'm interested in the movement of the ice edge as the pack shrinks...
41. ### Binomial Distribution and the Classical Definition of Probability

I am facing problems while comparing the results of solving a problem individually using both the concept of Binomial Distribution of Probabilities and the Classical Definition of Probability. Let me formulate the problem first: "The probability that a pen manufactured by a company will be...
42. ### Probabilities of Getting 50 Tails in 100 Coin Tosses Using Binomial Distribution

Homework Statement You throw a coin a 100 times, what's the probability of getting 50 tails? Homework Equations The Attempt at a Solution We have n=100 , p=1/2, q=1/2 and k=50 we substitute in the first equation we get: P= 100!/ (50! * 50!) * (1/2)^100 The factorials are not simple to...
43. ### Binomial distribution with dependent trials?

Hi to you all! I need your help with following problem: String with n characters is given. For each character in string there is probability p that it is wrong. Now you take a sliding window of length k, k<= n, that slides over that string. For the given parameters p,k and n one must must...
44. ### Probability Problem (maybe on Negative Binomial Distribution)

The following problem is from "Probability and Statistics in Engineering - Hines, Montgomery" A potential customer enters an automobile dealership every hour. The probability of a salesperson concluding a transaction is 0.10. She is determined to keep working until she has sold three cars...
45. ### Parameter space for the negative binomial distribution

Homework Statement For the negative binomial distribution, with r known, describe the natural parameter space Homework Equations the pmf for the negative binomial distribution with parameters r and p can be 1) P(X=x|r,p)= \binom {x-1}{r-1}p^{r}(1-p)^{x-r} where x=r,r+1,... , or 2)...
46. ### Self review: Statistics - Binomial Distribution

Homework Statement The Binomial Distribution - already developed by Jacob Bernoulli (in 1713), et alii, before Abraham de Moivre (1667-1754 CE), et alii, developed the Normal Distribution as an approximation for it (id est, the Binomial Distribution) - gives the discrete probability...
47. ### Why Do I Need to Multiply Probabilities in a Binomial Distribution?

please refer to the second line of solution, since we only concerned about the probability of getting number (5) , then why can't I just just say P=(5/6)^5 , why should I times =(5/6)^5 with (1/6)^2 ?
48. ### Stat mech and binomial distribution

Homework Statement Suppose that particles of two different species, A and B, can be chosen with probability p_A and p_B, respectively. What would be the probability p(N_A;N) that N_A out of N particles are of type A? The Attempt at a Solution I figured this would correspond to a binomial...
49. ### Binomial Distribution: Finding the number of trials

Homework Statement Question: Find the number of trials needed to be 90% sure of at least three or more success, given that probability of one success is 0.2 Homework Equations N/A The Attempt at a Solution My initial attempt at the problem was finding the probability of at least...
50. ### Binomial Distribution: Average & Probability of ≥1 Success

The average if the binomial distribution with probability k for succes is simply: <> = Nk So this means that if <> = 1 the distribution function must be peaked around 1. In general when is it a good approximation (i.e. when is the function peaked sufficiently narrow) to say that the...