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.

Probability is a numerical quantity which lies in the interval $[0, 1]$. According to the Bayesian view, it represents the belief about an event or preposition and it is interpreted as how likely it is for an event to occur, or of how likely it is for a proposition to be true. According to the frequentist view, it represents the relative frequency of a favourable event with respect to the sample space. Use this tag for basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantities. For questions about the theoretical footing of probability (especially using ), please ask under instead. For questions about specific probability distributions, please use .

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Cox proof of product rule - step explanation

I'm going through "Probability Theory - The logic of science" written by E.T. Jaynes and I have a problem with one step on page 27/28 in the proof of the product rule. The idea here is that we have a function $F$ for which: $$(AB|C)=F[(B|C),…
Marek G.
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Seemingly similar but different probability games

Burger King is currently running its "family food" game in which each piece can be modeled as a scratch off game where exactly one of three slots is a winner and you may only scratch one slot as your guess. As I was standing in line the other day I…
hackartist
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Normalized vector of Gaussian variables is uniformly distributed on the sphere

I have seen in various places the following claim: Let $X_1$, $X_2$, $\cdots$, $X_n \sim \mathcal{N}(0, 1)$ and be independent. Then, the vector $$ X = \left(\frac{X_1}{Z}, \frac{X_2}{Z}, \cdots, \frac{X_n}{Z}\right) $$ is a uniform random vector on…
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Probability that a clumsy boy eats $k$ out of 20 candies

A week or two (or maybe more) ago, the following question was posted and then deleted just as I was getting to the end of my solution. Unfortunately I have now forgotten what my solution was going to be. I don't think the OP reposted it, so here is…
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Expected number of tosses to get 3 consecutive Heads

I have a fair coin. What is the expected number of tosses to get three Heads in a row? I have looked at similar past questions such as Expected Number of Coin Tosses to Get Five Consecutive Heads but I find the proof there is at the intuitive, not…
RandomGuy
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PDF of the difference between two independent beta random variables

I am having trouble deriving the distribution of the difference of two beta random variables and would like some help verifying the steps I have taken. In particular calculating the bounds. Say I have $X_1\sim\text{Beta}(a_1,b_1)$ and…
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Biased coin with a $3/4$ chance to land on the side it was before the flip

Consider a hypothetical coin (with two sides: heads and tails) that has a $3/4$ probability of landing on the side it was before the flip (meaning, if I flip it starting heads-up, then it will have an only $1/4$ probability of landing tails-up). If…
Seth Wyma
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The probability of one Gaussian larger than another.

For two Gaussian-distributed variables, $ Pr(X=x) = \frac{1}{\sqrt{2\pi}\sigma_0}e^{-\frac{(x-x_0)^2}{2\sigma_0^2}}$ and $ Pr(Y=y) = \frac{1}{\sqrt{2\pi}\sigma_1}e^{-\frac{(x-x_1)^2}{2\sigma_1^2}}$. What is probability of the case X > Y?
Strin
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Exponential is to Poisson as Normal is to ???

In a time series, if the gap between successive events follows an exponential distribution with PDF $\lambda e^{-\lambda}$, then a Poisson distribution with parameter $\lambda$ tells you the probability of finding 0, 1, 2, etc events in time frames…
mathcsguy
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Die that never rolls the same number consecutively

Suppose we have a "magic" die $[1-6]$ that never rolls the same number consecutively. That means you will never find the same number repeated in a row. Now let's suppose that we roll this die $1000$ times. How can I find the PDF, expected number of…
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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 get up to speed quickly. Can anyone recommend…
MattK
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From correlation coefficient to conditional probability

In the best-selling book Thinking Fast and Slow (p. 205), Daniel Kahneman (a Nobel Prize winner in Economics) makes the following claim: 'Suppose you consider many pairs of firms. The two firms in each pair are generally similar, but the CEO of one…
Richard Hevener
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Let $X,Y$ have the same distribution on common prob space, do they generate same $\sigma$-algebra?

So let $X,Y$ be real random variables on common probability space $(\Omega, \mathcal{F}, P)$, the measures on Borel $(\mathbb{R},\mathcal{B}_{\mathbb{R}})$ induced by $X$ and $Y$ are equal, that is for all $A \in \mathcal{B}_{\mathbb{R}}$. $$…
them
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Coin tosses until I'm out of money

The question I think is a simple one, but I've been unable to answer or find an answer for it yet: There's a simple game: if you flip heads you win a dollar (from the house), but if you flip tails you lose a dollar (to the house). If I start with n…
Kang Su
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Laplace transform of integrated geometric Brownian motion

Is there any closed form of the Laplace transform of an integrated geometric Brownian motion ? A geometric Brownian motion $X=(X_t)_{t \geq 0}$ satisifies $dX_t = \sigma X_t \, dW_t$ where $W=(W_t)_{t \geq 0}$ denotes a Brownian motion and the…
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