# Questions tagged [probability]

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 (probability-theory) instead. For questions about specific probability distributions, please use (probability-distributions).

65,704 questions
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### Odds of drawing any 3 identical cards from 27-card deck (9 unique x 3 copies each) in 9 draws

I have a 27-card deck with 3 copies of each of 9 distinct cards. If I'm drawing 9 cards, what are the odds that I draw all 3 copies of any one of the 9 possible options? [Also known as "What are the ...
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### What's the probability that picking $3$ coins from a bag containing all coins from $1$ cent through $2$ euro will yield a total of $80$ cents?

I recently ran into this interview question and am wondering if my solution is correct. The setting is We have a bag containing one coin of each type, i.e. we have one $1$ euro cent coin, one $2$ ...
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### Probability - Too simple but confusing - Why two different approaches for similar problems?

There are two questions and the approach seems to be different. a) Probability of choosing 3 queens from a deck of cards? Ans: 4/52 * 3/51 * 2/50 However, if the question is how many diamonds ...
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### How to estimate the parameters of a censored exponential mixture process.

I have a real world scenario that involves a process behaving as follows: there are two kinds of machines and when these machines fail, their recoveries follow exponential distributions. Let's say the ...
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### Trying to understand a probability question/concept

Discrete Probability: Four dice are thrown, what's the probability that... So I made a post and there were a lot of incorrect answers, and two people were able to provide the correct answers, but ...
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### What is the probability of right and left handedness given these probabilities?

Me and some friends were having a debate over one of the questions presented during a lecture. According to my one of friend, who attended the lecture, he said that the professor said the correct ...
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### A poker hand contains five cards. Find the probability that a poker hand can be…

d) one pair Since there are 13 ranks, we pick 1 rank. $\binom{13}{1}$, for each of these ways we want to pick 2 cards out of 4 that belong in this rank. So $\binom{13}{1} \binom{4}{2}$. Now we want 3 ...
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### Discrete Probability: Four dice are thrown, what's the probability that…

Four dice are thrown, what's the probability that: a) None of them fall higher than three? b) None of them fall higher than four? c) That four is the highest number thrown? So for a, I wanna think ...
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### Odds of having at least one pair in a 4-card hand

I'm trying the calculate the odds of having at least one pair in a 4-card hand. I made the calculations in 2 different ways but the math doesn't add up. odds of having at least 1 pair in a starting ...
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### Pdf of $X+Y+Z$ where $X,Y,Z$ are independent $U(0,1)$

This is my working out of the problem so far, I want to know if there is a more simpler way to solve this, or I would just be interested in other methods that one could use to solve a similar problem
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### Monty Hall problem: If contestant knows the door with goat…

I need to know the probability of this variation of the problem. Suppose that the contestant is a psychic and in some way he/she knows one and only one door with a goat. *The psychic can't know the ...
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### Calculate $E(X|X+Y)$ where $X,Y$ have expotential distribution with parameter 1

Calculate $E(X|X+Y)$ where $X,Y$ have expotential distribution with parameter 1. I calculated distribution of $X+Y$, which if I am not mistaken is $\frac{e^{-x-y}}{x+y}$, where $x+y>=0$, but I am ...
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### Convergence of the sum of iid scaled by $n^\alpha$

I am interested in the convergence of the sequence $\mathbb{P}(|X_1+...+X_n|/n^\alpha<z)$ where $z>0$, $\{X_n\}_n$ is an iid sequence with mean zero and finite variance. I can easily prove that ...
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### Compare two normal probabilities

Assume $X_1,X_2,X_3$ are IID $N(0,1)$ random variables. I want to show that $$P(X_1>0,X_1+X_2+X_3<0)>P(X_1+X_2>0,X_1+X_2+X_3<0)$$ I know I can show this by calcultating these two ...
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### Expressing conditional independence in terms of third conditionally independent variable

So we have three variables $A$, $B$, and $C$. We know that $A$ and $B$ are conditionally independent given $C$ (ie $P(A,B|C)=P(A|C)*P(B|C)$) How would I prove $P(C|A)=\frac{P(B|A)}{P(B|C)}$. My ...
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### Finding a constant in a probability function

A basket contains one blue ball. Firstly, a random number of white balls, X, is added to the basket. The probability function of X is: $P(x=k) = \frac{c}{k!}$ ($k = 0,1,2,...$) ($c$ is a constant). ...
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### How to use a probability tree?

Suppose a student who is about to take a multiple choice test has only learned 40% of the material covered by the exam. Thus, there is a 40% chance that she will know the answer to the question. ...
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### Conditional Independence, Decomposition

Is there some set of independence relations between three random variables $X$, $Y$ and $Z$ such that $P(Z \mid X, Y)$ = $P(Z \mid X) \cdot P(Z \mid Y)$? (I feel like there should be, but I can't find ...
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### Markov process intensity matrix

$X$ is a Markov process with state space $(1,2,3)$. How can I find the matrices of transition probabilities $P(t)$ if the generator is \begin{bmatrix}-2&2&0\\2&-4&2\\0&2&-2\end{...
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### Optimal Bet Game Strategy

I start with a certain amount of cash $C$ and I play a game where on each win, I double my money. The game is structured such that the first time I play, I have a 50% chance of winning. Let's say I ...
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### Bounding sub-Gaussian tail events by Gaussian tail events?

Background I have come across a problem where I want to derive a concentration inequality for sub-gaussian random variables. More precisely, I want to bound the spectrum of a certain empirical ...
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### Notation Question: $X \sim \mu$ in the definition of Coupling

My professor defined a coupling as the joint distribution of $(X, Y)$ such that $X \sim \mu$ and $Y \sim \nu$ for probability measures $\mu$ and $\nu$. I was not quite sure whether $X \sim \mu$ ...
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### Setting up expression for distance of random point to origin

Problem Statement: Suppose that the coordinates of a point $(X, Y)$ are such that $X ∼ U[0, 1)$ and $Y ∼ U[0, 1)$ Compute the probability that the distance from this point to the origin is ...
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### Kelly bet question

I stumbled upon Kelly bet during studies and decided to try simulating it. I chose a simple example where $b = 1$ and $p = 0.6$, which gave me the Kelly bet equal to $0.2$ of the wealth. I took $25$ ...