Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using [tag:measure theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

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3answers
2k views

What is the probability that the coin drawn is fair

A box contains $n$ number of coins, $m$ of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is $1/2$, while it is $2/3$ when a biased coin is tossed....
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0answers
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Probability question about change in vending machines; maybe markov chain?

Suppose there are vending machine that sells its goods for $3$. It's known that a third of the buyers use three coins of $1$, a third of the buyers use $2$ and $1$, and the last third use $5$. The ...
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probability question related to pattern in coin tossing [closed]

If I toss a fair coin $n$ times, calculate the probability that no pattern HHTHTHH occurs.
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proof of equivalent definitions of recurrent states for Markov chains

I am wondering if anyone could prove the following equivalent definition of recurrent/persistent state for Markov chains: 1) $P(X_n=i,n\ge1|X_0=i)=1$ 2) Let $T_{ij}=min\{n: X_n=j|X_0=i\}$, a state $i$...
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2answers
837 views

1-D Random walk expectation calculation

I need assistance with the following question: The particle starts from the origin on the number line . It makes n steps at total. At each step it jumps with the probability 1/2 to the origin and ...
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1answer
122 views

Probability homework question

There are 6 of x and 4 of y. A group of 5 is selected. What is the probability of there being 3x and 2 y? I completely forgot how to do this and my textbook doesn't have any information on it.
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Improper Random Variables

What is an Improper Random Variable? I know the definition in terms of the CDF like, F(∞) - F(-∞) <1. Could any one explain it more clearly, specifically I am looking for an example of an improper ...
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1answer
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Probability of never reach -1 for a 1D random walk of n steps

On the number line, starting from 0. There is a probability of $p$ of moving 1 unit to the positive direction, and $1-p$ of moving 1 unit to the negative direction. What is the probability the walk ...
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Estimating population size

Let's suppose there are $n$ real numbers $a_0 < ... < a_n$ uniformly selected from interval [0, 1). If one knows $k$ numbers on consecutive positions $a_i < ... < a_{i+k-1}$ how good is $(...
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how can we generate random numbers using skew normal distribution in multivariate case

how can we generate random numbers using skew normal distribution in multivariate case
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2answers
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What is the probability that the boy is telling the truth?

My professor and I have came to a disagreement. My problem is with question number 8. I am pretty sure that you have to use the Bayes'thm for the question. I have tried ONCE to convince him, but ...
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How to estimate failure probability from count until first failure?

What would be the formula to estimate the rate of failure of some test as a percentage chance of failure from the number of runs of the test until the first failure was seen? For example, considering ...
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Self-study resources for basic probability?

I am taking a Computer Science class soon that requires a solid knowledge of the basics of probability. I've only had minimal exposure to probability in classes I've taken in the past, so I need to ...
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1answer
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Consequences of choice of a seed for random number generating algorithm?

Background I am trying to do a reproducible scientific analysis. My conclusions are not dependent on the random number generator, but the RNG does change the results ~1% between runs. I would like to ...
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1answer
809 views

random walk on the nonnegative integers

Consider a simple random walk on the integers with a reflecting barrier at $0$. What is the expected time the walk spends in the set $\{0, \ldots, k\}$ in the first $n$ steps? I'm curious about how ...
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4answers
345 views

Does this probability distribution have a name?

We have a set of numbers, of size $m$. We are going to pick $a$ numbers with uniform probability from that set, with replacement. Let X be the random variable denoting the probability of having X of ...
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3answers
600 views

Probability of getting exactly $k$ of a kind in $n$ rolls of $m$-sided dice, where $k\leq n/2$

This question was inspired by this question dealing with finding the probability of rolling a $7$ of a kind with $10$ dice. Is there a general method for finding the chance of rolling a $k$ of a kind ...
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13answers
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Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet?

Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet." I have a (not very mathematically intelligent) friend who ...
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4answers
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Chance of $7$ of a kind with $10$ dice

How can I calculate the probability of shaking $7$ of a kind using $10$ six-sided dice?
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3answers
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Expectation of time integral

I've 2 questions: let be $W_s$ a standard Brownian motion: using Ito's formula show that $\left( W_t,\int_0^t W_sds \right)$ has a normal distribution; and calculate $ E\left[e^{W_t}e^{\int_0^t ...
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1answer
132 views

Probability value in a skewed data

A question goes like this: Given that there 2 questions out of 3 questions to pass an exam and follow are details of number of people who attended the questions correctly, what is the maximum number ...
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3answers
383 views

Geometric distrubution and expectancy

I have following problem: We throw a fair dice time after time.What is time expectancy to get first time 66 i.e two six in the row for the first time. I probably should use this http://en.wikipedia....
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3answers
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Bounding probability that sum of random variables deviates from expected value

I'm working on a problem that requires me to find an upper bound on the probability that the sum of independent draws from a random variable deviates from the expected value of that sum by more than a ...
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1answer
769 views

Gaussian Distributed Random Variables

I know the sum of independent Gaussian distributed random variables also has a Gaussian distribution. say we have a sequence of Gaussian random variables not necessarily independent $ X_0, X_1, X_2, \...
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5answers
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Odds of Winning the Lottery Using the Same Numbers Repeatedly Better/Worse?

Does the probability of winning the lottery differ between randomly generated numbers vs. selecting the same numbers every time? Specifically. I'm interested in a breakdown of the odds per number for ...
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3answers
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How to evaluate probability estimators with only external information?

Here's a problem that I have pondered over many times without ever coming to a satisfactory solution: Let's say that we have a series of random events: V(i) for I = 1 to n. Each of these events will ...
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1answer
217 views

Expected amount of number one can add before reach n

Given integer $n$ and real $0\leq d<1$, set $x := 0$, set $c := 0$. set $y := n-x$, if $y < 1-d$, we are done Randomly chose $l\in [1-d, \min(1+d,y)]$, set $x := x + l$, $c := c+1$. Go to 2. So ...
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4answers
640 views

3rd axiom of probability for discrete distribution

it might be a stupid question but I was discussing with a colleague when the 3rd axiom of probability (sigma additivity) is really needed. I argue that in the case of a discrete distribution, say a ...
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1answer
370 views

Probability of obtaining triangle when choosing $3$ points from $3\times3$ array

$9$ points are placed in a $3\times3$ array. If $3$ points are randomly selected, what is the probability that they are the vertices of a triangle?
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2answers
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How to measure if a sequence has a bimodal distribution

Inspired by the recent incident in which a professor was able to [detect that students in his class were cheating][1], I'm curious if there is a standard way to detect if a distribution is bimodal, or ...
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1answer
446 views

Hitting times of reversible markov chain with known steady state probabilites

Consider a reversible markov chain Xn whose steady state distribution is known, can we find the expected hitting time to a subset A of the states starting from some state i ? Additionally you can ...
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1answer
271 views

Random parking problem on a probability distribution

Rényi's Parking Constants comes up when one puts down unit length cars on a interval, such that the probability of covering any two interval is the same. Are there any published results when the ...
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2answers
186 views

series of random variables

3:30am and i was just looking at the law of large numbers and thought about the idea of a series of random variables and got myself very stuck. when we consider the series $X'_n = X_1 + ... X_n$ and ...
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1answer
627 views

Probability of Profitability

Imagine I am betting on the outcome of flipping $n$ biased coins $C_i$. I know exactly what the probability of each coin landing on heads is $P(H)$ and exactly the probability of each coin landing on ...
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2answers
266 views

Consequences of boy/girl probability disparity on population

In a population that grows, would a disparity in having a boy/girl probability cause the ratio of males to females tend to the same? e.g. if the probability of having boys was $.49$ and having a girl ...
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1answer
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Probability - Calculate chance total score will be over or under a score line

I am working on a model which predicts the total final score (sum of the scores of both teams) of certain sports matches. I have checked the predictions of the model against the actual scores of ...
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2answers
388 views

Probability of finding $m$ or more cells empty

I asked I question here trying to obtain clarification about how to follow a hint. In spite of the fine answers I received there, the hint doesn't look very helpful. I'd like to know a hint for the ...
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4answers
9k views

What is average distance from center of square to some point?

How can I calculate average distance from center of a square to points inside the square?
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3answers
702 views

2 slightly different situations in which 2 coins are tossed. Does the knowledge of an observer effect the probabilities of the outcomes?

Situation A: Once only, I toss 2 identical fair coins and don't look at the outcomes. A truthful observer looks at one of the coins and tells me that at least one of the coins is a head. Situation B:...
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1answer
88 views

How can I add random “fuzz” to this formula, so that the randomness range diminishes as values increase?

I'm developing an open-ended strategy game. I am using the following formula to calculate damage (In PHP 'cause I don't know MathJaX!): ...
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Random Variable Inequality

Doing a little reading over the break (The Probabilistic Method by Alon and Spencer); can't come up with the solution for this seemingly simple (and perhaps even a little surprising?) result: (A-S 1....
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8answers
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When the roulette has hit 5 reds why shouldn't I bet to black?

First some context, I'm not a mathematician, not even close (as you will soon see) I do grasp some things about it but in a need to know basis, so plain english answers are appreciated (too). I can't ...
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1answer
216 views

Counting all possible size 'm' sets of 'N' marbles under intersection and membership constraints

Consider the case where we have a bag of 'N' unique marbles, and we randomly select all 'S' possible sets of size 'm' from the bag (with replacement) under the following constraints: (1) - Any set of ...
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In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?

In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? I just dipped into a book, The Drunkard's Walk - How Randomness Rules Our Lives, ...
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2answers
125 views

Effects of condensing a random variable to only 2 possible values

$X$ is a random variable, which is not constant. $E[X]=0$. $E[X^4] \leq 2(E[X^2])^2$. Let $Y$ be given by: $P(Y=E[X|X \geq 0]) = P(X \geq 0)$ and $P(Y=E[X|X \lt 0]) = P(X \lt 0)$. Do we necessarily ...
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1answer
278 views

Probability of finding exactly $m$ cells empty

Another problem on combinatorics. This time I'm asking for a hint and if it is possible a general strategy when dealing with this kind of problems. Show without using the preceding results * that the ...
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Probability that a random permutation has no fixed point among the first $k$ elements

Is it true that $\frac1{n!} \int_0^\infty x^{n-k} (x-1)^k e^{-x}\,dx \approx e^{-k/n}$ when $k$ and $n$ are large integers with $k \le n$? This quantity is the probability that a random permutation ...
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1answer
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Going to the Movies!

I was looking at movie times today and was struck by the oddly-spaced showing times. For example, at the local Loew's Theater "Tron: Legacy 3D" (127 min.) is playing on two screens at the following ...
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3answers
722 views

Thought experiment for dice game

Two players in a dice game to see who can roll a total of 60 first, taking turns each rolling 2 dice. For one player, one die is 4 sided and one is 6 sided. Therefore the average roll for this ...
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3answers
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Generating random values from non-normal and correlated distributions

I have a random variable X that is a mixture of a binomial and two normals (see what the probability density function would look like (first chart)) and I have another random variable Y of similar ...