Mathematics Stack Exchange
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Questions tagged [probability]

For questions about probability. independence, total probability and conditional probability. For questions about the theoretical footing of probability use [tag:probability-theory]. For questions about specific probability distributions, use [tag:probability-distributions].

The probability that an event occurs is a number in the interval $[0, 1]$, which represents how likely the event is to happen. $0$ indicates it will never happen, $1$ indicates it will always happen.

For example, throwing two dice gives a total of $6$ five times out of thirty-six. We write $$P(X=6)=\frac{5}{36}$$.

Use this tag for basic questions about probability, independence, total probability and conditional probability.

For questions about the theory of probability, use instead. For questions about specific probability distributions, use .

105859 questions
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Average time to find a duplicate with pairwise independence

Assume a process that samples uniformly at random from the range $[1,\ldots,n]$. I am interested in the expected time to find a duplicate given only that the sampling process is pairwise independent. That is I would like to find the expected time…
user35671
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Genetics and probability (“other than” case)

(Moved from Bio SE due to the mathematical nature of the problem.) If a man and woman, both carriers of a autosomal recessive disorder (i.e. having genotype $Aa$), produce three children, what is the probability of one or more children having…
lightweaver
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Card Shuffling Mathematics

In the paper "Trailing the Dovetail Shuffle to its Lair", Bayer and Diaconis give a formula for showing how many times a deck of $N$ cards has to be riffle shuffled for the deck to be considered random. The formula they came up with is $\frac32…
William Hird
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Family of distributions preserved under summing or product of their random variables

I wonder what families of distributions can satisfy that the sum of their any two random variables still have a distribution in the same family? what families of distributions can satisfy that the product of their any two random variables still…
Tim
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If a fair six-sided die is rolled four times, in how many outcomes is the value of each roll at least as large as the value of the previous roll?

Suppose you roll a fair 6-sided die four times. Let C be the event that the value of each roll is at least as large as the value of the previous roll. What is the probability of C? I know that $$\omega = 6^4 = 1296$$ I also know that to get P(C), I…
rocketman
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Pairwise independence vs independence

Two fair dice are thrown. We have three events: A: The first die shows an odd number B: The second die shows an even number C: Both are odd or both are ven Show that $A,B,C$ are piecewise independent but not independent. My answer: $P(A) =…
Yakub
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Are $|X|$ and $\operatorname{sgn}(X)$ independent?

Let $X$ be a real valued random variable. Let $\operatorname{sgn}(x)$ be $1$ when $x>0$, $-1$ when $x<0$ and $0$ when $x=0$. Why are $|X|$ and $\operatorname{sgn}(X)$ independent, when the density function of $X$ is symmetric with respect to…
Tim
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r distinct balls in N boxes

If r distinct balls are distributed at random into N (N ≤ r) boxes, what is the probability that box 1 will receive exactly j balls ( 0 ≤ ≤ r)? my solution is [sample space] =$ N^r $ $$P=\frac{ 1}{N^r}\binom{r}{j}$$ I know there is something…
user311726
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Probability of winning a game between players A and B?

The following problem is from A First Course in Probability by Sheldon Ross, and it was assigned as homework by my professor. I was wondering if you guys could help me find a answer to the problem. $A$ and $B$ flip coins. $A$ starts and continues…
Birdman2246
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What is the probability that a randomly chosen positive three-digit integer is a multiple of $7$?

What is the probability that a randomly chosen positive three-digit integer is a multiple of $7$? Is my answer right?: $$\frac{100}{7} = 14 , \qquad \frac{999}{7} = 142$$ Then there are $142 - 14 = 128$ numbers that are multiples of $7$. Then the…
TheFermat
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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 this is a program that count how many random numbers…
Chao Xu
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Probability and game

This should be known as "gambler's ruin". In a game, at each step, you can win 1\$ or lose 1\$. Let $Z_i$ be a variable that can assume as values 1 or -1. Let $$ X_n=\sum_{i=0}^n Z_i . $$ Can you show me in details how to calculate $P(X_n \geq a)$…
Flast9
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Law of the Iterated Logarithm and Stopping Times

Let $(X_1,X_2,...)$ be i.i.d random variables with mean $0$ and variance $1$. By the Law of the Iterated Logarithm, for all $\epsilon >0$, \begin{equation} P\left[ \frac{1}{t}\sum_{i=1}^{t}X_i \geq (1+\epsilon)\sqrt{\frac{2 \log \log t}{t}}\text{…
Matt
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Probability of choosing two real numbers $a$ and $b$ from $[1,4]$ such that $ ab>4$.

What is the probability that when you pick two real numbers from the closed interval $[1,4]$, their product is greater than 4? I tried to solve it with integration but I couldn't get the right answer. And I think that this problem can be solved…
user265554
  • 2,853
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If a sequence of quadratic forms converges in probability and a random vector converges in distribution then $X_n^TQ_nX_n$ converges

If a sequence of quadratic forms converges in probability $Q_n\xrightarrow{P}Q$ and a random vector converges in distribution $X_n\xrightarrow{d}X$ then $X_n^TQ_nX_n\xrightarrow{d}X^TQX$. This is a statement from an online source in statistics. It…
Rodrigo
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