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Questions tagged [binomial-distribution]

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 yes/no experiments, each of which yields success with probability p. (Def: http://en.m.wikipedia.org/wiki/Binomial_distribution)

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Distribution of product of two binomial coefficients

I am considering the following distribution on $\ell \in \{0,1,\dots,k\}$ originated from the product of two (interlaced) binomial coefficients: $$\frac{\binom{\delta n}{\ell}\binom{(1-\delta)n}{k-\...
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Joint distribution of the sum of binomial variables

Suppose $Y_1, Y_2, ..., Y_m$ are independent binomial variables each comprising $n$ trials with success probability $p$. For any $Y_j$ define $I_j$ as, \begin{align} I_j = \begin{cases} 0 &...
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Statistic test - critical value

So my application is a little different but, I will try to use a slightly simpler example. Essentially what I am trying to calculate is that if I flip a coin $100$ times what number of flips is ...
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Why does the Binomial Theorem use combinations and not permutations for its coefficients?

I have been trying to understand the Binomial Theorem formula. I can see that it works. What I don’t understand is how or why using combinations finds the coefficients. What I mean is, isn’t each ...
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A limit of expected value of binomial function

Claim: Let $F$ be a cumulative distribution function with support $[0,1]$, and let $a$, $b$ be two given numbers such that $0\leq a< b\leq 1$. Consider the probability mass function of the binomial ...
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Reading Binomial Tables

While reading a table of cumulative binomial probabilities, if I need to find the probability of, for example, exactly 4 successful events happening and all the rest failures occurring, how would I go ...
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Calculate Binomial Probability for inequality

How could the binomial probability be calculated for a case of $P(X \geq 2)$ for a given value of $n$ and $p$ ?
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Binomial distribution of $P(x>8)$

how do I do binomial distribution of $P(x>8)$, given $X\sim B (10,0.3)$ lets say for $P(X\ge2) = 1- p(X=0) - P(x=1)$ why is $P(x>8) = P(x=9) + P(x=10)$, how did this come about and why i cant ...
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Help understanding the Binomial Distribution for nonindependent events?

Apologies if this seems like a basic question, but I haven't been able to find a clear answer in my textbook or online. The original problem; Please determine the probability of finding exactly 3 ...
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How to use Binomial distribution to figure out probability of X number of success per N times?

Let's say a success happens $120$ times per minute, also $120$ times per $60$ seconds ON AVERAGE. And this means $2$ times per second on average. And we assume we can use Poisson distribution to ...
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A random variable $X$ is number of boys out of $n$ children. Calculate $\operatorname{Var}(2X-n)$

Let a random variable $X$ be the number of boys out of $n$ children. The probability to have a boy or a girl is $0.5$. Calculate $V(2X-n)$. I know that $Var(2X-n)=4V(X)$. $\mathbb{P}(X=k)={1\over 2^...
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Expected value of number of throws of a dice to get element $1$ four times

Here is the question: Find the expected value and variance of the number of times one must throw a dice until the element $1$ has been obtained $4$ times. My attempt: The minimum number of throws ...
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method of moments estimator for binomial distribution

I am presented with the following homework problem: Consider N independent random variables having identical binomial distributions with the parameters $\theta$ and $n = 3$. If $n_{0}$ of them take ...
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When will $f(i):=\binom{2k-1}{i}\Big((1-p)^i(1+p)^{2k-1-i}-(1+p)^i(1-p)^{2k-1-i} \Big)$ attain maximum?

When will $$f(i):=\binom{2k-1}{i}\Big((1-p)^i(1+p)^{2k-1-i}-(1+p)^i(1-p)^{2k-1-i} \Big)$$ attain maximum among $i=0,1,\dots,k-1$, for very large positive integer $k$, and $p\in (0,1)$ with $p=\Omega(...
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Determine P(X > Y) with given X and Y

A new treatment for a disease is being tested, to see whether it is better than the standard treatment. The existing treatment is effective on 50% of patients. It is believed initially that there is a ...
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Two players alternate flipping a coin until the result is head. How to derive that the probability for the first player to win is $2/3$? [duplicate]

Two players, $A$ and $B$, alternately and independently flip a coin and the first player to obtain a head wins. Player $A$ flips first. What is the probability that $A$ wins? Official answer: ...
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Poisson and Binomial RVs

Compute $E[e^{tX}]$ as a function of t ∈ R when (1) X is Poisson λ, (2) X is Binomial n, p. [Hint: you may find the Binomial Theorem to be useful]. The PMF of Poisson is $$ p_x(k)=e^{-λ}λ^k/k! $$ ...
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Assume a distribution of test scores following N(75,81), and a score greater than or equal to 60 means “pass”.

Assume a distribution of test scores following N(75,81), and a score greater than or equal to 60 means "pass" (with a full score of 100). What proportion of the students pass? In a random sample of ...
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Isolating a needle from a haystack of binomials distributions

Let $0<p<q<1$ two fixed and known parameters. Define $\mathcal{P}$ (resp. $\mathcal{Q}$) the binomial distributions of parameters $p,R$ (resp. ($q,R$) for $R\in \mathbb{N}$. Suppose that I ...
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Price and Probability Calculation on Producing Items

I'm hoping someone can help me out here. Product A can be produced from Product B at a 50% chance. Product B can be produced from Product D at a 25% chance and a cost of \$1400 success, and \$700 ...
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Binomial Distribution Probabilities - Need help with multiple requirements for k

Could someone please check if the procedure for part a) is correct and I need a bit of help in regards to part b). I don't have the answer to refer to. Thank you for your time in advance! An airport ...
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Binomial distribution with mean and standard deviation

Can anyone help me with homework? I can`t catch a basic idea of the next exercise: $$ X\thicksim Bin(20,0.2)\\P(|X-\mu|\le\sigma)=? $$ Can anyone at least explain how it opens,please?
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Derivative of binomial distribution

Consider the function $$B(M) = LM-I\cdot\sum_{x=n+1}^{M}{ M \choose x}p^x(1-p)^{M-x}(x-n)$$ What is the first derivative of $B(M)$?
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Binomial Distribution with Traffic Accidents

Here's the problem: The probability that there is no accident at a certain busy intersection is 95% on any given day, independently of the other days. Today was accident free. Find the probability ...
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If $X$ has a binomial distribution with parameters $n$ and $p$, calculate $\textbf{E}[(1+X)^{-1}]$ [duplicate]

If $X$ has a binomial distribution with parameters $n$ and $p$, show that \begin{align*} \textbf{E}\left(\frac{1}{1+X}\right) = \frac{1-(1-p)^{n+1}}{(n+1)p} \end{align*} MY ATTEMPT Since $X\sim\...
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Why does the conditional probability of having chosen a biased coin given an unusual result decrease with a higher number of coins

Let's say $1$ out of three coins is biased and lands on tails with probability $p>1/2$ We choose a coin randomly from the three coins and throw that coin $10$ times. Given that it lands on tails $...
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Fast(er) way of computing the cumulative binomial probability?

While revising for my probability test, I saw this question from one of the previous exams: You flip a fair coin 100 times. What is the probability that you have less than 45 heads? My question is a ...
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Two coins, one experiment: what is the probability that the coin lands on heads on exactly $7$ of the $10$ flips?

When coin $1$ is flipped, it lands on heads with probability $0.4$; when coin $2$ is flipped, it lands on heads with probability $0.7$. One of these coins is randomly chosen and flipped $10$ times. (...
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Showing monotonicity for ratio of binomial pmf and tail cdf

I'm interested in showing for $X\sim\text{Bin}(n,p)$, $p\in(0,1)$ that when $x\geq np$, $$ \frac{P(X=x)}{P(X\geq x)}\leq \frac{P(X=x+1)}{P(X\geq x+1)} $$ I've verified using numerical simulations, but ...
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Finding probability of success from cumulative binomial distribution

Given the binomial PDF $P(X = j) = \frac{n!}{(n-j)!x!}p^{j}(1-p)^{n-j}$ where $n$ is the number of trials, $j$ is the number of successes, $p$ is the probability of success, is there some simple ...
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Bimodal Distributions and Binomial Simulations

I have a bimodal distribution for a singular simulation and I need to find a model that fits this. I was thinking a Binomial Model, but is that the best option? Also how do I then simulate 100 trials ...
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When should I apply binomial distribution equation?

I would start by describing the question: Assuming A-Z are letters assigned with different probabilities of appearance. We have a target pattern in letters: A--K-M, six letters in length with three ...
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Expectation and variance of the number of elements of a random non-empty set selected from a finite power set

Let $S$ denote a finite set of cardinality $|S| = N$. Select randomly a non-empty subset of $S$. Let $X$ indicate the number of items belonging to this subset. (a) Describe the probability mass ...
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Donkey hitting his head on the same stone - a question about probability

I am practising for an exam that has a section on introductory probability theory, we have covered basic Markov chains and conditional probability + binomial distribution, not much more. In these ...
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What is the probability of getting two heads twice in $4$ tosses of two coins?

What is the probability of getting two heads twice in $4$ tosses of two coins? My Attempt: No of trials $(n)=4$ Probability of success in one trial $(p)=\dfrac {1}{2}$ Probability of failure $(q)=\...
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Distribution for $n$, the number of draws required to reach a target value

I draw with replacement from a pot containing 10 balls: 7 red, 3 white. If I draw a red ball I score 2 points, and if I draw a white ball I score 1 point. I stop drawing when I reach 10 or more points....
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Probability of selecting more than x of a color given distribution?

If you have 2000 candies distributed uniformly over 5 colors, what is the probability of getting more than 300 blue candies? I was thinking of doing complementary counting, so $$ 1-\sum_{i=0}^{300} ...
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Statistics : Hypothesis Testing [closed]

I'm back with another question I can't solve. For this question, I was able to get part of the answer; nk$^{n-1}$(1-k), but I can't seem to see where you'd get the k$^n$ part. Questions Answers
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Confidence Interval question for amount of experiments one should do.

Before posing this question, the lecture notes I am reading discussed games, probability, the binomial distribution and central limit theorem. It usually assumes some form of game when it asks ...
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Probability of football player scoring 2 goals, 3+ goals or anytime given his prob to score if goal is being scored

Assuming that I have a football match, and each team has 11 players. I know for any given goal the probability of scoring for each player (for simplicity these probs remain constant during the game). ...
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What is the probability of having 10 or more crashes in the first 1000 rides?

You have recently invested in an electric skateboard. You always wear a helmet and padding to protect yourself (of course), but are worried that at some point you'll take a tumble and get scratched up....
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Standardize binomial variable for non-constant meta-population - binomial z-scores

Let's say I'm doing a meta-analysis of a general experimental protocol that has been applied to some experiments (with 1,0 type outcomes) across a variety of experiment-type sub-groups. I want to test ...
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Why p$\frac{\partial }{\partial P} $ is the mean of binomial distribution function?

Why p$\frac{\partial }{\partial P}[ {p+q}]^n $ is the mean of binomial distribution function? I know the mean should be $\sum np(n) $ but why p$\frac{\partial }{\partial P} [{p+q}]^n$ of the binomial ...
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Cumulative conditional probability

The question I have can be boiled down to this example problem: There is a bag with 10 balls, 8 of which are white, 1 of which is red, and 1 of which is blue. In n trials, where you pull a ball, ...
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Probability of a single trial within binomial experiment vs. stand-alone bernoulli experiment

When a flip a coin several times, each throw is independent from another. In other words, my coin does not know what came out previous time. So, each next flip the result is unpredictable and random. ...
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rolling a dice multiple times: when the distribution is binomial and when it is not

I suppose we roll a 6-sided dice three times. A probability to to obtain two times number 3 can be framed as binomial because I can define as "p" a probability to obtain 3 and "1-p" the remaining ...
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Taking limit of a sequence inside discrete distribution function

Suppose that there exists a positive, integer sequence $\{c_k\}$, satisfying $c_k\geq c^*$, such that $c_k\to c^*$. Further suppose that $X$ is a binomial random variable with size $n$ and parameter $...
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Bean machine (Galton board) explanation

I am confused by the marked part of the following explanation (see below; https://en.wikipedia.org/wiki/Bean_machine). Suppose the bead bounces to the right twice and to the left also twice. It will ...
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Approximating binomial with normal distribution: probability and density values are practically the same?

In discrete distribution when we plot PMF the Y axis is probability. In continuous distribution when we plot PDF the Y axis is density (probability is the area under the curve). So, we learn that ...
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The Probability that Eric Reid (NFL Player) is being targeted with 6 drug Test in 11 weeks.

Ok, I thought this was interesting and wanted to see if anyone could figure it out cause the media can't. Here is the situation, Eric Reed is an NFL player, He has been in the NFL for 11 weeks ...