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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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Hi guys. I am abit confused as to how the percentages are calculated of busting when hitting on specific Blackjack hand values.

Blackjack probability Not sure how the percentages are derived and what probability formula is being used. Please help
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Problem about the throwing of three dices

I have this problem: If three dice are thrown, what is the probability that the sum of the numbers obtained is 6? The only solution that occurs to me is to look for all the possible cases by ...
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Probability of Winning a Game Question

An urn has 9 white balls and 11 black balls. You draw a ball from the urn, note its color, and put it back in the urn. If it is white, you win 5 cents from your opponent; if black, you lose 5 cents to ...
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Understanding probability word problems

I'm currently having trouble translating word problems into their respective probabilities. For example: Suppose that we that we have three cups: A, B, and C. A contains two red marbles, B ...
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probability that sum of geometric random variables is odd

Let $X$ and $Y$ be independent geometric random variables with the same parameter $p$. What is the probability that $X+Y$ is odd? Based on the provided solutions, I can follow through until this part ...
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2answers
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Die roll and coin flip - Bayes' theorem

I am thinking to the question posed here: Die roll and coin flip. "Suppose I roll a 4-sided die, then flip a fair coin a number of times corresponding to the die roll. Given that i got three heads on ...
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1answer
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Median of a continuous r.v.

Let X be a continuous r.v., with pdf $f_X(x) = kx(1-x), 0 < x <1$ I evaluated k = 6 and found the cdf $F_x(x) = 3x^2 - 2x^3$, but then I am asked to find the median. The equation $F_x(x) = 1/2$ ...
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Probability problem about a deck of cards.

Some cards were thrown away from a standard pack of $52$ cards. However all four aces were made sure to be kept among the remaining cards. Four cards were then selected at random from these ...
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Probabilities of choosing a red ball:Case 1) 10 distinct balls,8 red and 2 black Case 2) 10balls ,8 identical red and 2 identical black

I know the probability in both cases evaluate out to be 8/10. I want to know the intuition behind solving the probability of choosing a red ball when there are 8 identical red balls and 2 identical ...
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1answer
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At least probability of two students

I have been trying to view this problem for a while and I have come unstuck, because of the at least at least part. Calculate the probability that at least one of two students A and B, who ...
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1answer
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What is the probability that in a group of n people chosen at random, there are at least two born in the same month of the year?

So I'm working on a probability problem: In Exercise 19 assume it is equally likely that a person is born in any given month of the year. b) What is the probability that in a group of $n$ ...
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Linking Markov Chain with Renewal Process

GIVEN: $X_0,X_1,...$ irreducible, recurrent Markov chain with transition matrix $P$ Starting state $X_0=x$ $g(m)=P\{X_m=y\}$ for some fixed state $y$ I know that the renewal process is $g(m)=b(m)+\...
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What is the probability that the monkey will type the phrase “Call me Ishmael”?

Random events are independent events. Consider a typical computer keyboard with 82 keys. And a monkey typing on this keyboard, at random. The output of the typing would look like: jw9:.2wb0288q 1nej@...
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1answer
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Problem to calculate a marginal function in probability

I have a problem in probability. I have $f(x, y) = \frac 14 \cos(y) $, if $x$ is between $0$ and $\pi$, and if$ y$ is between $-\frac x2$ and $\frac x2$. I have to calculate $f(y)$. I calculated ...
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Generating the number of success of an bernoulli process

I want to simulate the process of rolling $n$ dice and counting the dice which show a 6. The pmf for this formulation is $$ \text{P}\left(X = k\right) = \binom{n}{k}\left(\frac{5}{6}\right)^{n-k}\...
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Probability of having picked every item from the set at least once after n turns, while picking 3 per turn

Let's say I have a set of 100 items. Each turn, I pick three items at random, note which ones I've picked, and put them back. What is the probability I've picked every item at least once after $n$ ...
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Uncorrelatedness for random elements of finitely generated groups?

Suppose $G$ is a finitely generated group, $A$ is its finite set of generators. Lets denote the metric induced by the Cayley graph $Cay(G, A)$ on $G$ as $d$. Suppose $\{X_i\}_{n = 0}^\infty$ is a ...
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Sampling from Geometric distribution in constant time

I would like to know if there is any method to sample form the Geometric distribution in constant time without using $log$ which can be hard to approximate. Thanks.
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When does $E[f(X_i)]=E[f(X_j)], i\neq j$?

Suppose we have random variables $X_1, \dots, X_N$, with joint probability distribution $F_{X_1,\dots,X_N}$. Under what conditions does the following equality holds? $$E[f(X_i)]=E[f(X_j)],\ \ i\neq ...
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2answers
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Of ten fifty-peso bills, three are counterfeit. Six bills are chosen.

Of ten fifty-peso bills, three are counterfeit. Six bills are chosen at random. What is the probability that all counterfeit bills are chosen? I tried solving for the number of ways the six bills ...
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Calculating probabilities from sigmoids

The problem is that I would like a user to rate a genre on a scale from 1 to 100. This input would go into the function as $x and return a number from the sigmoid function. The reason for the sigmoid ...
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Drawing without replacement: why is the order of draw irrelevant?

I am trying to wrap my head around this problem: Daniel randomly chooses balls from the group of $6$ red and $4$ green. What is the probability that he picks $2$ red and $3$ green if balls are ...
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1answer
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Probability distribution of a moving particle

I am having a issue with the wording of this question. Find the probability of the following. The velocity $v$ of a randomly selected particle, whose distribution obeys the probability density ...
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1answer
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Under what conditions $g(X_n,Y_n)\overset{d}{\rightarrow}g(X,Y)$?

Under what conditions, given $X_n\overset{d}{\rightarrow}X$ and $Y_n\overset{d}{\rightarrow}Y$ $\Rightarrow$ $g(X_n,Y_n)\overset{d}{\rightarrow}g(X,Y)$ I know that we can't apply the continuous ...
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A Casino game called Winno consists of playing 10 bets. [on hold]

A gambler wins a bet if he or she plays a "6" or a prime number from the outcome of a fair die. Each bet costs GHS10.00 and a win yields GHS15.00. However, if the gambler loses a bet, he pays ...
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1answer
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Probability as a Measure of Belief and Conditional Probabilities

I came across the following question in my textbook and it made me think about how to calculate conditional probabilities when we consider probability as a measure of belief. Joe is 80 percent ...
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Distribution of life time of a serial circuit with bulbs

Assume that we have a serial circuit with three bulbs. Each bulb's life time is exponentially distributed: $$f_{bulb}(t) =\left\{ \begin{aligned} &\lambda e^{-\lambda t} & t \ge 0\\ &0 &...
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estimate Markov chain mean transition time

Assume a continuous time Markov chain which is run through in one direction and finally absorbed at the last state $1 \rightarrow 2 \rightarrow 3 \rightarrow ... \rightarrow n $ The transition ...
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2answers
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Probability of not Drawing 5 Kings

A special deck of cards just contains the face cards (jack, queen, king) of each suit. What is the probability that you draw: A hand of five cards in which you have no kings? My Attempt: $(1-4/12)(...
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1answer
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Questions regarding mutual independence of events [on hold]

Have a few questions regarding mutual independence: If I have a set of events $A_1, A_2, …A_n$ that are all pairwise independent, it is possible that the events may not be mutually independent? If I ...
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1answer
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How to determine $E[X^3]$ given $E[X^2]$ and $E[X]$

Let $X$ be a discrete random variable with $E[X] = 2$ and $E[X^2] = 4$. Find $E[(2 + X)^3]$. When I proceed to $E[2+4X+8X^2+X^3]$ I don't know how to calculate $E[X^3]$.
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Find the P.M.F. of $X_{1}+X_{2}$ given $(X_1,X_2,X_3,X_4) \sim \text{Mult}(n,4,p_1,p_2,p_3,p_4)$

I can find the marginal P.M.F.s of $X_1$ and $X_2$ but then I am lost on how to convolute the two PMF's into one PMF. I should be using convolution formulas right? because that is the only way I can ...
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2answers
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Calculating probability of 8 letter String [on hold]

Computer generate 8 string password with letters a,b,c,d,e. Each letter is chosen independently and with equal liklihood of being selected. What is the probability that the string has 8 of the same ...
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4answers
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What is the intersection of an event with itself?

The probability of the intersection of two event is: $P(A \cap B) = P(A)P(B)$ If the two events are the same, i.e $P(A \cap A) = P(A)P(A) = P(A)^2$ However, the logic tells us that the probability ...
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1answer
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Maximum Likelihood Estimator for Non Absolutely Continuous Distributions

Let $\theta\in[0,1]$ be a parameter. Suppose that $Y$ is a random variable that takes value $\theta$ with probability $1/2$ and is uniformly distributed on $[0,1]$ with probability $1/2$. What is ...
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Probability problem of win a game

I have this problem about probability: If in a bingo, the probability for each game to win a prize is 0.2, and in each game, a single prize is drawn. What is the probability that a person ...
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2answers
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PMF of function of random variables

Let X be a Geom($\frac{1}{2}$) random variable, and define Y=$X^{-1}$ What is the p.m.f. of Y ? attempt: pmf of a Geom RV in general form is $p(1-p)^{k-1}$ There is this similar question, not ...
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1answer
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How Would You Calculate Your Edge With These Variables?

Say we are playing a casino game where we have there potential outcomes, and 3 probabilities associated for each one, all which sum to 1. They are: Win $1.00 (80%) Lose $4.00 (17%) Lose $1.00 (3%) ...
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Poisson Process example from Durret's Probability textbook

I'm struggling to understand some examples related to the following theorem in Durret's Probability: Theory and Examples. The theore is: For each $n$, let $X_{n,m}, 1 \leq m \leq n$ be independent ...
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2answers
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Shortcut to finding the distribution of a specific random variable

Question: A dice is rolled 3 times. Let X denote the maximum of the three values rolled. What is the distribution of X (that is, P[X = x] for x = 1,2,3,4,6)? You can leave your final answer in terms ...
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Calculate Mean from the Moment Generating Function ( m.g.f / mgf) of Y = 2X +3

The question is as follows: No calculators. Let X be a random variable with moment generating function $M_{x}(t) = \frac{e^{(e^t-1)}}{2e^{-t} -1} \;\;\;\;\;for \;\; \;t<log(2)$ given Y = 2X +3 ...
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1answer
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Prove $\lim_{x\to\infty} x . [1 - F(x)] = 0$ [duplicate]

I have no idea how to proceed with proving this. If $X$ is a continuous random variable, $P(X > 0) = 1$, $E(X)$ is defined and $F(x)$ is the CDF, then prove $\lim_{x\to\infty} x . [1 - F(x)] = 0$
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1answer
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$P(X = n) = pq^{n-1}$ where p, q > 0 and p + q = 1. Find Var(X) using generating function.

$P(X = n) = pq^{n-1}$ where $p, q > 0$ and $p + q = 1$. Find ${\tt Var}(X)$ using generating function. First I found $E(X)$: $$\sum_{n=1}^\infty q^n = 1/(1-q) - 1 = q/(1-q)$$ then differentiate ...
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2answers
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Density function of an exponential distribution

Let $X$ be a random variable with an exponential distribution with $\lambda=1$ and $Y=2X$. What is the density function of $f_y$? I know that $$f_x =\begin{cases} e^{-x} & 0\leq x\leq\...
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1answer
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Asymptotically tight bounds $P \left[1-\frac{1}{n} \le \frac{\sum_{i=1}^n Z_i^2 }{n} \le 1+\frac{1}{n} \right]$

I am looking for assymptotically tight bounds on \begin{align} P \left[1-\frac{1}{n} \le \frac{\sum_{i=1}^n Z_i^2 }{n} \le 1+\frac{1}{n} \right]=P \left[ \left| E[Z^2] - \frac{\sum_{i=1}^n Z_i^2 }{n} ...
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conditional expected value of mean preserving spread

My first post on here, and my math skills are more than a little rusty. I have a simple question for you: assume $Y$ is a mean preserving spread of $X$. I'm trying to find a general proof for the ...
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Reverse engineering distributions

Suppose I am given a measurable function $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and a probability distribution $\mathbb{P}$ on the Borel or Lebesgue sigma algebra of $\mathbb{R}^n$. Assume that the ...
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1answer
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Does it make sense to compare coefficient of variance between samples with different sample size?

I have two samples with different sample sizes. The difference is quite large: one has sample size of 10 and one has sample size of 200. Two samples are same type of data but are collected from two ...
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2answers
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Coin game between A and B

Consider the following simple game. In a single round, Player A tosses a fair coin, and then Player B tosses a fair coin. Two rounds are played. The winner is the player with the larger number of ...
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Prove periodicity is a class property

Prove that if state $i$ in a class has period $p$ then all states in that class have period $p$. The proof is given on this answer is this: One way to define the period of state $i$ is as the ...