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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-3
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0answers
11 views

Factories and parts

Two factories produce similar parts for the same company. Factory 1 produces 2000 pieces of which 10% are defective. The factory 2 produces 4000 pieces of which 7.5% are defective. One part is ...
5
votes
4answers
43 views

Two dice thrown, one comes up 6

If my friend throws two dice, and covers them up, but I see that one of them was a 6, what's the probability that they were both 6s given this knowledge? I'm under the impression that the answer is ...
1
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1answer
11 views

Prove that the discrete time martingale can be represented by $E (Y_{n +1} \mid F_n) = 0$ if $Y_{n +1} = X_{n +1}-X_n$, for $n = 0,1, \ldots $

Prove that the discrete time martingale can be represented by $E (Y_{n +1} \mid F_n) = 0$ if $Y_{n +1} = X_{n +1}-X_n$, for $n = 0,1, \ldots $ I want to use the sequence $(y_n)$ called "martingale ...
0
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1answer
28 views

Probability and Statistics homework

A country doctor has 40 patients. Every day he visits 3 of them. Jason is one of the doctor's patients. What is the probability that Jason will be visited by the doctor in any particular week? ...
0
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2answers
22 views

Probability of a draw given 15 pairs of objects

Given 15 pairs of objects, what is the probability that you will draw exactly one of your desired object if you were to take 3 out of the 30 total objects? If possible, work this out by not using the ...
0
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1answer
9 views

Autocorrelation of a Wiener Process proof

Given a Wiener process X, how do I prove this? $R_x(s,t) = E[X(s)X(t)] = min(s,t)$ There seems to be a trick with dividing to two cases of $s<t$ and $s>t$, but I can't figure out how this ...
0
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0answers
15 views

Covariance and correlation of correlated random variables

Let $X, Y, Z$ be random variables having mean 0, variance 1. Let $\rho_1, \rho_2, \rho_3$ be the correlation between $X$ and $Y$, $Y$ and $Z$, and $X$ and $Z$, respectively. Show that $\rho_3 \geq ...
0
votes
1answer
12 views

How to find Reliability of a rectangular distribution function?

Assume that the failing of a device is equally probable within an interval [a,b] such that the fault density is: f(x) = {1/b-a if a<= t <= b ...
0
votes
1answer
35 views

Calculate the Probability in competition

The committee RAM competition knows from experience that the probability of successfully Contest is 0.95 for the student who has grade "very good" in BAC test , 0.5 one who has 'Good' in ...
0
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3answers
18 views

How many ways to fill six broadcast slots for 3 adertisements which are to be shown twice

A television director is scheduling a certain sponsor’s commercials for an upcoming broadcast. There are six slots available for commercials. In how many ways may the director schedule the commercials ...
0
votes
1answer
18 views

What is the probability of choosing 5 random students whose (individual) grade is higher than 1149 points?

I know this isn't that hard, but I have been looking and I don't know how to solve it. The number of students whose grade is higher than 1149 is 44, and the total of students is 135. If the question ...
0
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2answers
28 views

Expectation of a function of weighted sum

Set of non negative weights $w_j$, set of non negative i.i.d. random variables $X_j$ and $f(y)$ is a decreasing nonnegative function in $y$. I want to claim that if $\sum w_i<\sum w^{\prime}_i$ ...
2
votes
1answer
21 views

Probability question with Geometric random variable

Sir Lancelot and Sir Galahad are doing a shoot out, in which they try to shoot each other while shooting in the same time at each other. The probability of Sir Lancelot to hit Sir Galahad is 0.5 and ...
0
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3answers
62 views

Birthday paradox, huge numbers

Pick x random "birthdays", say $10^9$. What are the chance of a collision, given $2^{160}$ possible "days"? I'm trying to estimate the collision rate of sha1 hashes, but the calculation is too big ...
0
votes
2answers
32 views

Non-Probabilistic Argument for Divergence of the Simple Random Walk

The simple random walk is one starting at $0$ with steps of $-1$ and $1$ with equal probability. Is there a proof not involving (too much) probability - preferably number-theoretic - of why this walk ...
0
votes
1answer
30 views

After $n$ sticks are broken into two parts each, they are joined again randomly. Find the probability of them being joined in a certain way

Each of $n$ sticks are broken into a longer and a shorter part. Out of these $2n$ parts, $n$ sticks are formed again by joining any 2 parts randomly. Find the probability that a) The parts will be ...
3
votes
3answers
32 views

Probability of having 4 aces after taking turns to pick cards

I've started to learn probability and I get stuck with the following problem: My friend and I are playing a card game with 36 unique cards. There are four suits (diamonds, heart, clubs and spades), ...
1
vote
1answer
28 views

Probabilities of calling with Cell phones

Cell phones perform transfers as they move from cell to cell. During a call, a phone can make zero transfers ($H0$), a transfer ($H1$) or more of a transfer ($H2$). Additionally, each call is ¨long¨ ...
0
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2answers
25 views

Schwartz Inequality (probability) - first step in proof

I'm trying to understand the Schwartz Inequality for random variables, which states $$(E[XY])^2 \leq E[X^2]E[Y^2] $$ The solution states that we can assume $E[Y^2] \neq 0$ because if this were the ...
1
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4answers
83 views

The probability that after repeated random drawing from an urn, all balls left in the urn will be red

Problem An urn contains $p$ red and $q$ green balls. Balls are drawn one by one till balls left in the urn are all red. Prove that the probability of this event is $\dfrac {p}{p+q}$. Please note that ...
0
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1answer
27 views
0
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1answer
35 views

A simple problem on probability

Suppose we have a train running on a railroad from $A$ to $B$. The railroad is N Km long from the point $A$ to the point $B$ and the speed of the train is $v$ $Km/h$. We have two situations: in the ...
1
vote
1answer
31 views

Odds/Probability

I don't know the proper terminology but I would like to get a primer on odds and probability. If the odds are set at 1 in 500 what is the probability of winning on the first try (0.02%?), after 500 ...
0
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0answers
16 views

Exact probability distribution for hitting time of simple random walk

Consider simple random walk on the line starting from the site $y \in \mathbb{N}$. With probability $p$ the walker moves to the right and with probability $1-p$ to the left. Call $\tau$ the first time ...
1
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5answers
54 views

3 distinct numbers are selected from $\{1,\ldots,9\}$. What is the probability that 9 is selected?

3 distinct digits are randomly selected from the set of nine digits [1-9]. What is the probability that 9 is selected. I thought that the probability should be (1/9) + (1/8) + (1/7) since you have to ...
0
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0answers
28 views

Probability of guessing Date of Birth

Is it possible to give the probability of guessing the Date of Birth of a complete stranger i.e. to the exact year, day and month? If so, what would that be?
0
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0answers
38 views

Probability of getting head in coin flip [duplicate]

Suppose a football match is going to be started. The referee should flip a coin to give the ball to one of the football teams. In front of the two captains the referee flips the coin two times and ...
0
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0answers
19 views

Approximating the probability of an event by finite-dimensional distributions

Let $(X(t))_{t\ge 0}$ be a stochastic process on $\mathbb{R}^d$, say an Ito diffusion (with continuous sample paths). Let $A\subset \mathbb{R}^d$ be a measurable set and $t>0$. Does the following ...
0
votes
1answer
14 views

infer the initial state from draws

I went through binomial distribution and Chi-square test etc and got confused further. This question might be very basic and simple. I have three states (Combination of two colors, both has equal ...
1
vote
0answers
15 views

On the geometry description of the GSR riffle shuffle model

In 1992 Diaconis and Bayer announced their famous result which is now a well-known folklore: Seven shuffling is enough to randomize a deck of cards. One of the key ingredients in their proof is that ...
1
vote
1answer
29 views

E[X] and E[X^2] with Conditional Expectation

$\newcommand{\E}{\operatorname{\mathbb E}}$ $\newcommand{\Var}{\operatorname{\mathbb Var}}$ If $\E[X] = {^1\!/\!_3}(\E[X\mid Y=1] + \E[X\mid Y=2] + \E[X\mid Y=3]) = 10$ Where $\E[X|Y=1] = 2,\; ...
0
votes
3answers
50 views

Should I pick the higher dice?

Assuming I start with $n$ dice that have been rolled once, is it beneficial to choose the higher dice when I roll less than $n$ dice again (assuming I want a high roll)? In some board games, dice ...
0
votes
1answer
36 views

Show that $X_n \stackrel{d}{\longrightarrow} 0$ iff $\{\varphi_n(t)\}$ converges to 1 in some neighbourhood of $t=0$.

$X_n$ is a sequence of random variables, and $\{\varphi_n(t)\}$ is the corresponding sequence of characteristic functions. Show that $X_n \stackrel{d}{\longrightarrow} 0$ iff $\{\varphi_n(t)\}$ ...
0
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1answer
16 views

Probability inequality exchanging sum with cardinality

Let $P_{XY}$ be the joint probability distribution of discrete random variables $X$, $Y$. Then I would like to prove the following inequality: $$ \sum_{y}\max_xP_{XY}(x,y)\leq |Y|\max_xP_X(x) $$ ...
2
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2answers
21 views

Determining the maximum % below average

Is there a way to determine the maximum percentage of values that fall below the average in a given sample? How would someone go about this? How does this relate to what Markov's inequality and ...
0
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0answers
24 views

Statistics: How would I correlate many variables to a few coefficients?

I'm trying to predict the "strength" vs "tempertature" and "time" curves of some chemical compounds as a function of the concentrations of their component substances. I have 20 different substances. ...
0
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0answers
14 views

Is there a bound on largest eigenvalue for covariance matrix of discrete random variable?

I have a random variable $Z=(Z_1,\ldots,Z_p)$. Each component can take values in {-1,0,1}. Is there a way to bound the largest eigenvalue of Cov(Z)? Actually, I have a latent multinormal variable ...
0
votes
2answers
27 views

What does “Choose N ~ Poisson(ξ), Choose θ ~ Dir ( α )” mean in the context of Latent Dirichlet Allocation

I'm reading http://machinelearning.wustl.edu/mlpapers/paper_files/BleiNJ03.pdf and trying to understand the notation and concepts behind LDA, in order to implement it myself. I've followed some ...
4
votes
2answers
521 views

Outcome of rolling a fair die 6 times

I'm failing to understand how to come to the answer to this question. If you roll a fair die six times, what is the probability that the numbers recorded are $1$, $2$, $3$, $4$, $5$, and $6$ in any ...
8
votes
3answers
447 views

Does 0% chance mean impossible? [duplicate]

Suppose we pick a random real number between 0 and 1 and call it $x$. There are $2^{\aleph_0}$ possible values, so the chance of picking any specific number (such as $x$) in that range is 0. But in ...
4
votes
2answers
73 views

Probability of Sum of Independent Events Exceeding a Value

Suppose I have $n$ random number generators. Once an hour, on the hour, each one generates a random real number $x_k$ such that $0 \le x_k \lt \infty$. Each generator produces its values according to ...
2
votes
2answers
40 views

Neyman-Pearson lemma. Doubt on the text of the lemma

In my book: $\mathbf{X}=(X_1,\ldots,X_n)$ $f(\mathbf{x})$ is the joint density, where $f$ is either $f_0 \text{ or } f_1$. Suppose we want to test $H_0: f=f_0$ or $H_1: f=f_1$. The test, whose test ...
0
votes
1answer
20 views

limiting joint distribution

Let $X_n\xrightarrow[d]{}N(0,\sigma^2_x)$ and $Y_n\xrightarrow[d]{}N(0,\sigma^2_y)$. $X_n, Y_n$ are not independent. Can I say that $\left( \begin{array} {} X_n \\ Y_n \end{array} ...
0
votes
0answers
24 views

conditional on zero probability event [duplicate]

let $ ( \Omega, F, p ) $ be a probability space, A, B be events with $p(B) = 0$, then what is the definition of P(A|B)? I wonder it for a long time since it happens all the time when using continuous ...
0
votes
1answer
24 views

Probability static area is greater than variable area

I have a rectangular section of constant height and length and I choose a random starting point anywhere along its length. From the randomly chose starting point, I first add ...
1
vote
1answer
38 views

Roll a fair die. You lose as many dollars as the number of pips (spots on the dies) that are showing…

Then you toss a fair coin as many times as the number of pips. For each heads, you win \$20; for each tails, you lose \$1. Let X = total amount you win (or lose if $X<0$). What is $E(X)$? My ...
0
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0answers
48 views

The probability of random permutation leaving the sequence almost unchanged

So let's say I have $52$ completely different kinds of arenas that get shuffled. What are the chances of getting the exact same sequence if only one arena can be out of order? For example, you could ...
2
votes
4answers
23 views

Probability of picking some color combinations of socks

A box contains $3$ yellow socks, $4$ blue socks, $1$ orange sock, and $2$ green socks. What is the probability of picking $2$ blue socks at the same time? What is the probability of picking $1$ green ...
1
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2answers
41 views

Conditional Expectation Discrete and Continuous

Find $E[X]$ and $Var[X]$ So for the expectation so far I got that: $$E[X] = E[X|N=n]P(N=n) = \large\frac{n+1}{\lambda} \frac{\lambda^{n}}{n!}e^{-\lambda}$$ but for conditioning on both a discrete ...
1
vote
3answers
44 views

Can this conditional probability be answered using Bayesian Theorem (or at all) with the information given

I have a conditional probability problem I'm unsure can be answered given the information I have - as such I'm unsure if Bayesian Theorem is the way to answer it, or if the answer is staring at me in ...