Binomial Definition and 640 Threads

Homework Statement 50 students live in a dormitory. The parking lot has the capacity for 30 cars. If each student has a car with probability 12 (independently from other students), what is the probability that there won't be enough parking spaces for all the cars? Homework Equations P(A) =...

Homework Statement For reference, this is the image setting up the problem. "A wireless sensor grid consists of 21×11=231 sensor nodes that are located at points (i,j) in the plane such that i∈{0,1,⋯,20} and j∈{0,1,2,⋯,10} as shown in Figure 2.1. The sensor node located at point (0,0) needs...

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...

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

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...

Homework Statement Calculate {-3 \choose 0}, {-3 \choose 1}, {-3 \choose 2} Homework Equations In case of integer ##n## and ##k## { n \choose k}=\frac{n!}{k!(n-k)!}=\frac{n(n-1)(n-2)...(n-k+1)}{k!} The Attempt at a Solution I am not sure how to calculate this. Any idea?[/B]

Hello all, I have this equation \sum_{k=\lceil \frac{n}{2}\rceil}^n{n\choose k}P^k\left(1-P\right)^{n-k}=\epsilon and I want to find P as a function of epsilon and n. Can I do that? If so, then how? Note: \epsilon<10^{-3} if it helps for any possible approximation. Thanks

Homework Statement Prove that ∑nj=0(-1)j(nCj)=0Homework Equations Definition of binomial theorem. The Attempt at a Solution If n∈ℕ and 0≤ j < n then 0=∑nj=0(-1)j(nCj) We know that if a,b∈ℝ and n∈ℕ then (a+b)n=∑nj=0(nCj)(an-jbj) Let a=1 and b= -1 so that 0=(1+(-1))n=∑nj=0(nCj)(1n-j(-1)j)...

Homework Statement Show that the binomial coefficients ## \binom {-1}{n}=(-1)^n## Homework Equations ##\binom{n}{k}=\frac{n!}{(n-k)!k!} \\ ## The Attempt at a Solution ##-1!=(-1)\cdot 1! \\ -1!=-1 \\ -2!=(-1)^2 \cdot 2! \\ -n!=(-1)^n \cdot n!\\ \mbox{for n=0} \\...

The below image shows a portion of my current Analytical Mechanics textbook. My inquiry is how is the binomial theorem used to get from eq. 3.4.5a ⇒ 3.4.5b ? Thanks in advance

Homework Statement ##\sum\limits_{r=0}^n\frac{1}{^nC_r}=a##. Then find the value of $$\sum\sum\limits_{0\le i<j\le n}(\frac{i}{^nC_i}+\frac{j}{^nC_j})$$ Homework Equations I have used two equations which I derived myself. This is the first one. The second one is: 3. The Attempt at a...

Homework Statement Four players play a board game which requires them to take it in turns to throw two fair dice. Each player throws the two dice once in each round. When a double is thrown the player moves forward six squares. Otherwise the player moves forward one square Homework Equations...

Any hints for this : 1- (11C1/2.3 ).2^2 + (11C2/3.4 ). 2^3 ...so on up to 12 terms .

Homework Statement So I am working on an Electric Potential problem. There is a point P that is located on top of this rod ( this rod is aligned horizontally & is length L). I've solved this problem and got an answer. I want to find when y>>L using Binomial Approximation except I am quite...

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...

Homework Statement The wait time (after a scheduled arrival time) in minutes for a train to arrive is Uniformly distributed over the interval [0,12]. You observe the wait time for the next 95 trains to arrive. Assume wait times are independent. Part a) What is the approximate probability (to 2...

Hi all, I have an apparently simple equation. I copy here its Mathematica code: Sum[(p/(1 - p))^s*(q/(1 - q))^s*Binomial[n, s]*(Binomial[m - 1, s]*(p*q*(m + n) + (2*m - 1)*(-p - q + 1))), {s, 0, n}] == Sum[(p/(1 - p))^s*(q/(1 - q))^s*Binomial[n, s]*((-(-p - q + 1))*Binomial[m - 2, s] +...

The binomial series ##(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!} x^2 + ...## only converges for ##|x| < 1## right? Is it true that writing ##(1 + x)^n## differently (i.e. ##x^n (1 + \frac{1}{x})^n##) extends the validity of this series to include values of ##x## such that ##|x| > 1##?

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...

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...

I'm using Pascal's (n choose k) method for calculating the coefficients of the terms of a binomial expansion. However, if the exponent is a negative integer, how can one use this method, seeing as factorials for negative integers are undefined. For example, how could one determine the...

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...

Homework Statement The number of claims that an insurance company receives per week is a random variable with the Poisson distribution with parameter λ. The probability that a claim will be accepted as genuine is p, and is independent of other claims. a) What is the probability that no claim...

Homework Statement Sorry if this is a dumb question, but say you have 1/(1-x) This is the form of the geometric series, and is simply, sum of, from n = 0 to infiniti, X^n. I am also trying to think in terms of Binomial Series (i.e. 1 + px + p(p-1)x/2!...p(p-1)(p-2)(p-(n-1) / n!). 1/(1-x) is...

Homework Statement Let p be a prime, k be positive integer, and m ∈ {1, 2, 3, ..., pk-1}. Without using Lucas' theorem, prove that p divides \binom{p^k}{m}. Homework Equations The definition of the binomial coefficients: \binom{a}{b} = \frac{a!}{b! (a-b)!} The Attempt at a Solution I've...

Hey guys, can you help me with the proofing of E(X) = r/p and Var(X) = rq/p^2

Homework Statement Consider the expansion (ax2 + bx + c)n = ∑(r=0 to r=2n) Ar xr------------------(1) , where Ar is real ∀ 0 ≤ r ≤ 2n Replacing x by c/(ax) and using the property ∑(r=0 to r=2n) Tr = ∑(r=0 to r=2n) T2n-r , we get (ax2 + bx + c)n = ∑(r=0 to r=2n) Br xr...

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...

Homework Statement The integer next to (√3 + 1 )^2n is -- (n is a natural number) Ans: Divisible by 2^(n+1) Homework EquationsThe Attempt at a Solution (√3 + 1 )^2n will have an integral and a fractional part. So, I + f = (√3 + 1 )^2n (√3 - 1 )^2n will always be fractional as (√3 - 1) < 1 So...

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...

Homework Statement Find the coefficient of x^n in the expansion of ( 1 + x/1! + x^2/2! + x^3/3! + ... + x^n/n! )^2 . Homework EquationsThe Attempt at a Solution At first glance, this looks like the polynomial form of e^x, but the expansion of e^x goes to infinity, so any use of that seems...

Homework Statement Use the binomial expansion (1± x)n = 1± nx + (n(n-1)/2) x2 ±... to show that the value of g is altered by approximately Δg ≈ -2g(Δr/rE) at a height Δr above the Earth's surface, where rE is the radius of the Earth, as long as Δr<<rE Homework Equations g=GM/r2 The Attempt...

Homework Statement This involve testing population proportion. (either small or big sample) this question is done by my lecturer by using binomial , i am wondering could it be done using normal distribution? because the np is 20(0.45) = 9 which is greater than 5 ...

This the only question I'm having issues with. It may be a binomial distribution or poissm, not really sure. If an airplane has 224 seats and the no show rate of passengers with reservations is .09 how many reservations should the airline book such that the probability of not enough seats for...

Screenshot by Lightshot The translation in binom coefficent of 4th and 10th are mathching each other. Find the member which doesn't have x in it. I understand all of it but the part where (n up n-3)=(n up 9) I just don't understand how they got 12 here

In solved binominal form (4x+3)^n has two members x^4 and x^3 whose binomial coefficients are equal. I'm kinda good in solving binomial coefficient, but I never stumbled to something like this

Homework Statement Dear Mentors and Helpers, here's the question: Find the range of validity for (1 + 3x/2)^(-1) and (1 + 1/(3x))^(-1). Homework EquationsThe Attempt at a Solution For the first binomial series: -1 < 3x/2 < 1 -2 < 3x < 2 (multiply 2 throughout) -2/3 < x < 2/3 (divide by 3...

Background: I am trying to write a program for Buckingham Pi Groups. I need to find a way to list all the input varialbes as different sets. For example if I have 4 variables [V D p u] and I want to distribute them 3 ways I get 4 sets. Number of Sets = Binomial(Number of Variables...

Homework Statement Prove that, if x is so small that x^6 and higher powers of x may be neglected, then \frac{e^{2x}-1}{e^{2x}+1}\approx x-\frac{x^3}{3}+\frac{2x^5}{15} Homework EquationsThe Attempt at a Solution [/B] \\...

Homework Statement Find the coefficient of x^3 in the binomial expansion of (2/x - 3x^4)^12 Homework EquationsThe Attempt at a Solution Expanding this out would take too long and I cannot use a calculator to find the coefficient I know the formula for the expansion summation (12 choose k)...

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...

Homework Statement The binomial expansion of (1+x)^n, n is a positive integer, may be written in the form (1+x)^{n} = 1+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+...c_{r}x^{r}+... Show that , if c_{s-1}, c_{s} and c_{s+1} are in arithmetic progression then (n-2s)^{2} =n+2 Homework Equations The Attempt...

In the binomial theorem (x+x')^n = a summation of terms. Is there any way I could express (x+x')^^n in a summation?

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...

I've been thinking about this for some time. Now I'm coming on here in the hope of getting some help; Prove that an+3 + (a + 1)2n+3 is divisible by a2 + a + 1. I can't quite remember the restrictions on n, though I'd imagine it'd be "for all real n ≥ -1" or something similar. Thanks in...

Use the binomial approximation to derive the following: A) γ=1+.5(β^2) B)1/γ=1-.5(β^2) C)1-(1/γ)=.5(^2) I know the approximation is 1+(.5β^2)+(3/8)β^4+... A) is self explanatory but not sure how to derive B) and C)

Homework Statement From an old exam: Show that \begin{equation*} \sum_{0 \leq 2k \leq n} \binom{n}{2k}2^k = 0 (3) \text{ iff } n = 2 (4). \end{equation*} By ##a = b (k)## I mean that ##a## is congruent to ##b## modulo ##k##. Homework Equations Binomial theorem: ## (a + b)^m =...

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...

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)...

Imo, this problem is crazy hard. Homework Statement Let X have the negative binomial distribution with pmf: f_X(x) = \binom{r+x-1}{x}p^{r}(1-p)^{x}, x=0.1.2..., where 0<p<1 and r is a positive integer. (a) Calculate the mgf (moment generating function) of X. (b) Define a new...