# Questions tagged [probability]

For basic questions about probability and the questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities. 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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12 views

### All of Statistics example 2.11

Two people take turns trying to sink a basketball into a net. Person 1 succeeds with probability 1/3 while person 2 succeeds with probability 1/4. What is the probability that person 1 succeeds before ...
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### Does the probability of a coin toss always stay 1/2 no matter the previous tosses

Assumption 1: The coin is unbiased Assumption 2: The tosses are independent Suppose i toss a coin 3 times and each time it came up heads then what is the probability that the next toss would be heads. ...
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### Understanding the Probability Outcomes

A eight sided dice is rolled thrice and the first roll was an odd number. If the selected roll in random is odd what is the probability? Total number of possible outcomes when a die is rolled thrice = ...
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### Given $Y_i$ with pdf $f(y) = my^{m-1}/n^m, 0<y\leq n, 1\leq i \leq k$, prove that $-k\ln (\frac{Y_{(k)}}{n})$ is Gamma$(1,1/m)$

Given $Y_i$ with pdf $f(y) = my^{m-1}/n^m, m>0, n>0, 0<y\leq n, 1\leq i \leq k$, prove that $-k\ln (\frac{Y_{(k)}}{n})$ is Gamma$(1,1/m)$. I tried finding the CDF of $f$. I used that to find ...
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### please help me. I have probability generating function as follows, how can get the mixture distribution of $X$ from it?

$$G_x (s)=\frac{s \cdot (1-q(1-α+α \cdot s))}{(1-q \cdot s)(1-α+α \cdot s)}$$ Or, in another way, how can I say that a random variable follows a certain distribution with a certain probability and ...
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### Conditional variance of the sum and difference of two independent variables?

Suppose that $X$ and $Y$ are independent with mean 0, variance 1. Suppose $A = X+Y \\ B = X-Y$ I read that $Cov(A, B)=Var(X)-Var(Y) = 0$. Can I say anything about the terms in the conditional ...
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### Probability of getting a specific sequence of length $4$ in $10$ coin tosses

This is more of a thinking question maybe, hope that's ok. Suppose I toss a coin $10$ times. What is the probability that within these 10 tosses I get the sequence THHT. My attempt: If I have 10 coin ...
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### Two players rolling a $20$-sided die; player B can re-roll; how to decide when to reroll

This is somewhat related to my earlier question What is the probability that player A rolls a larger number if player B is allowed to re-roll (20-sided die)? and somewhat related to 30 sided die and ...
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### How do you write the law of total expectation for a conditional expectation?

I would like to compute $E[N|D]$, expectation of some random variable $N$ given that event $D$ occurs. In the outcome space, there are a set of disjoint events $B_1, B_2, B_3$. How can I apply the law ...
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### Understanding the mean function for time series analysis

I am studying time series and came across the Mean function which the textbook defines as: $$\mu_{xt}=E(X_t)=\int_{-\infty}^{\infty}xf_t(x)dx$$ I don't understand what this function does. I looked ...
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### Probability of catching covid - statistical model

I have been thinking about this approximate model of what is the probability of you catching covid based on the assumption how many people you were in contact with that would transmit it to you, if ...
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### What is the probability that player A rolls a larger number if player B is allowed to re-roll (20-sided die)?

The problem statement is: 2 players roll a 20-sided die. What is the probability that player A rolls a larger number if player B is allowed to re-roll a single time? The question is a bit ambiguous, ...
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### Given joint pdf, how can I find $\operatorname{Var}(X)$ without using the marginal distribution of $X$?

$(X,Y)$ has joint pdf $\frac{1}{y}$ for $0<x<y<1$. I would usually use the marginal pdf to get the expected value. But the question doesn't let me use the marginal distribution of $X$. I ...
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### Two players until one player wins three games in a row. Each player will win with probability $\frac{1}2$. How many games will they play?

QUESTION: Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with ...
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### Ball Selection without replacement

A box contains $2$ white balls, $3$ red balls and $4$ black balls. Now I have to select $2$ balls one white and one red from the box without replacement. I'm confused between whether the probability ...
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### How can I model a difference in probabilities?

I am trying to optimize the strategy to choose while taking a penalty kick. I have identified two strategies- keeper independent and keeper dependent and I am trying to find the best one using a ...
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### Calculating expected value of $X$ with the density function $f(x)=16xe^{-4x}$

Suppose, $X$ be a random variable with probability density funciton, $$f(x) = \begin{cases} 16xe^{-4x}, & x \geq 0; \\ 0, & \text{otherwise} \end{cases}$$ (source) I tried to find the ...
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### Convergence of Uniform random variables

Let $U$ be a Uniform[0,1] random variable. If we consider $[nU]$ where [] is the greatest integer function, then we know that $[nU]$ is a discrete uniform $\{1,2,...,n-1\}$ random variable. We know a ...
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### Bayes' theorem and law of total probability with CDFs

Suppose $X$ has Gamma(2, λ) distribution, and the conditional distribution of $Y$ given $X = x$ is uniform on $(0, x).$ Find the joint density function of $X$ and $Y,$ the marginal density function of ...
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### Gamma Random Variable Confusion

I know that the gamma random variable can be thought of as an extension to the exponential random variable. The exponential random variable measures the time it takes for one event to appear, while ...
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### Box containing 4 black and 6 white marbles.

Four marbles are drawn from the box at random without replacement. Calculate the probability that at least two white marbles are drawn given that at least one of each colour is drawn. I have ...
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### Calculating the population size given a probability

I'm not too sure how to even start this question. Could someone explain how to solve this question :) Question: The minimum number of times that a fair coin can be tossed so that the probability of ...
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### Finding Expected Value of Conditional Poisson Distribution

Consider a random variable X ~ Poisson (1). Namely, $P(x=k) = \frac{e^{-1}}{k!}$ , k=0,1,2,... I'm trying to solve for $\mathbb{E}\{X|X\geq 1\}$. My approach: Given that $X\geq1$ then we know that at ...
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### Application of law of total probability

Consider the discrete random variables $X,Z,W$ with supports $\mathcal{X}, \mathcal{Z}, \mathcal{W}$ respectively. Let $\mathcal{W}\equiv \{w_1,w_2,w_3\}$. Take any $x\in \mathcal{X}, z\in \mathcal{Z}$...
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### Markov chain magic trick (Kruskal Count)

I am trying to understand why this works. We have a Markov chain here with 10 states which are the numbers 1,..,10. There is a none zero probability to pass between any two states, so we can show that ...
When the density $f_{X, Y}$ is not defined for independent random variables $X, Y$, is it possible to say anything beyond \begin{align} Var(X + Y) &= Var(X) + Var(Y) + 2 \int_{\Omega} (X - E(X))(Y-...