Tagged Questions

This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under (...

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Bayes theorem and conditional probability

I have a problem like this: Seventy-eight percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, ...
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Conditional Probability and life expectancy

In a population of 100,000 females, 89.835% can expect to live to age 60, while 57.062% can expect to live to age 80. Given that a woman is 60, what is the probability that she lives to age 80? Using ...
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Is it true that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$?

I was wondering if we can show that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$ in general? Here $X_1$ and $X_2$ are independent but may not follow the same distribution. Any hint is much ...
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What does it mean if $cov(f(x1), f(x2))$ is positive in the context of LHS sampling?

If cov(f(x1),f(x2)) is positive, does that mean f is close to symmetric along x1 and x2? I am struggling to put this into understandable terms. Edit: The context is equation 6 in this paper: http://...
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Calculate probabilies based on given probability distribution

A mail-order company business has six telephone lines. Let $X$ denote the number of lines in use at a specified time. Suppose the pmf of $X$ is as given in the accompanying table \begin{array}{r|...
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Example of a set and monotone class where monotone class is not a $\sigma$-algebra?

What is an example of a set $X$ and a monotone class $\mathcal{M}$ consisting of subsets of $X$ such that $\emptyset \in \mathcal{M}$, $X \in \mathcal{M}$, but $\mathcal{M}$ is not a $\sigma$-algebra?
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Figuring out probability of two random events both happening

So here's the problem: The table below shows the distribution of education level attained by US residents based on data collected during the 2010 American Community Survey: ...
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Exact Probability of reducibility of Bivariate Polynomials

I am considering polynomials of the form $$P(x,y)= \sum_{k=0}^n\sum_{l=0}^n a_{k,l}x^{k}y^{l}$$ where $n \in \mathbb{N}$. The coefficients $a_{k,l}$ are considered to be randomly generated from the ...
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How to Calculate the “Drop Off” of a Set

So I have never taken a formal class of statistics and this is likely just a case of me not knowing the right name for what I am looking for. Nonetheless, say I have a set of numbers in descending ...
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How to calculate t-value, given degrees of freedom and $\alpha$.

While solving problems, we can look up physical t-tables or use a statistical analysis software like R to calculate t-values. But how do we actually calculate these values ? What is the algorithm ...
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probability of getting an erdos number once published [closed]

Can I know that I don't have an erdos number once I published, what the probability is of getting an erdos number with "random" coauthors or can I formulate the probability of having a finite erdos ...
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distribution and density of maximum minus element

I am a bit rusty in probability, and for a project I am studying the random variable $Z = \max(X_1, \ldots, X_n) - X_i, i = 1, \ldots, n$ where the $X_i$ are positive independent random variables. In ...
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The Spacing of $e$ and $\pi$ Segments Within the Decimal Expansion of $\pi$

I discovered something seemingly very improbable today when I was searching for segments of $e$ and $\pi$ within the decimal expansion of $\pi$. I searched for $314159265$ and found it starts at the ...
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What are the odds of flipping a coin 100 times and seeing HHHHT? [on hold]

What are the odds of flipping a coin 100 times and seeing exactly four consecutive heads? Any more than four heads in a row, such as "HHHHH" would not be considered a string of four consecutive heads. ...
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Conditional probability using set notation [on hold]

Got this wrong on a quiz and i don't have the answers. Need to figure this out for a test coming up. \begin{align} P(A) &= 0.75 \\ P(B\mid A) &= 0.9 \\ P(B\mid A^c) &= 0.8 \\ P(C\mid A\...
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When to stop pumping up balloons?

Yesterday I acted as a volunteer in a psychology/neurology experiment where one of the trials consisted of playing a computer game in which you had to click the mouse to pump up a balloon. For each ...
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Suppose X and Y have joint density f (x, y) = 2 for 0 < y < x < 1. Find P (X − Y > z). [on hold]

Suppose X and Y have joint density f (x, y) = 2 for 0 < y < x < 1. Find P (X − Y > z). Solution is (1-z)^2
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Probability Sum

A purchasing agent must decide to accept or reject an incoming shipment of machine parts. The agent wishes to do either of the following: a1: Accept the shipment a2: Reject the shipment The fraction ...
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Find the limit of the probability of uniform random variable?

Let $X_1 ,X_2 ,X_3 ,…$ be a sequence of i.i.d. uniform $(0,1)$ random variables. Then, calculate the value of $$\lim_{n\to \infty}P(-\ln(1-X_1)-\ln(1-X_2)-\cdots-\ln(1-X_n)\geq n)?$$ My work: Since ...
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A binomial sum identity

Let \begin{align*} f(n, r, \pi, k) &= \sum_{z=0}^{n}\sum_{s=0}^{r}\binom{z}{s}\binom{n}{z}\binom{n-z}{r-s}(-1)^{r+s}\left(\frac{\pi}{1-\pi}\right)^{r/2-s}\pi^{z}(1-\pi)^{n-z}z^k \end{align*} I am ...
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Expected score from threshold with number deletions

We play a game where a sequence of $n$ numbers is drawn uniformly from $[0,1]$, and we need to set a threshold $0\leq a\leq 1$. For every number that is at least our threshold, we get $a$ points but ...
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Random probability

So the story is my friend was playing Runescape and he was trying to get an item drop that had a ${1\over 128}$ drop rate so on average every $128$ monsters he slays one of the items will drop.he ...
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recursive definition for two mutually exclusive events [on hold]

How do we write recursive definitions for two mutually exclusive events ? Can anyone explain with some examples as how do we come up with solutions in case of exclusive events ? SO finally i add ...
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Probabilistic Method/Model for Traffic Flow

Context: Given a network system or a traffic system with some value related to the system. Question: Which probabilistic methods, model, distributions are used frequently to predict a event (for ...
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Sub sigma algebra and probability spaces — definition

I am reading this book and I am a bit lost with the definitions because they are not provided and I can't seem to find it online: Let $L_2(\Omega,A,P)$ be a probability space such that $f \in L_2$ ...
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Length of left stick [on hold]

Break the stick into 3 pieces and what would be the expected length of left stick? I need answer for this to verify my answer. Can somebody give me the answer? My thoughts: My answer is 1/4. First ...
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determine the distribution of the random variable $Y=\Sigma_{k=1}^{\infty}kX_k$

Fix $p \in (0,1)$ and consider independent Poisson random variables $X_k$, $k \geq 1$ with $\mathbb E[X_k]=\frac{p^k}{k}$. Verify that the sum $\Sigma_{k=1}^{\infty}kX_k$ converges with probability ...
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Creating unusual probabilities with a single dice, using the minimal number of expected rolls

Problem I want to create an 'event' with probability of $\frac{1}{7}$ with a single dice as efficiently as possible (to roll the dice as little as possible). To give you some better understanding of ...
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Variance: logical/mathematical meaning [duplicate]

$$\operatorname {Var} (X)=\operatorname {E} \left[(X-\mu )^{2}\right]$$ Is the formula of variance. But if you think of it, you can assume that square was introduced just that something other than ...
Proving Pascal's Rule : ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$
I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$. I suppose I could use the counting rules in probability, perhaps combination= \${{n}...