Mathematics Stack Exchange
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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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Is there a deeper meaning to the identity ${{\sin x}\over x} = \prod_{k=1}^{\infty} \cos\left({x\over{2^k}}\right)$?

Is there any deeper meaning to trigonometric identity $${{\sin x}\over x} = \prod_{k=1}^{\infty} \cos\left({x\over{2^k}}\right)$$ beyond it corresponding to characteristic functions of i.i.d. random variables?
user166854
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From a pack of 52 playing cards, three are drawn at random. Find the probability of drawing a king, a queen and jack.

A simple question but the solution is confusing me. The answer I obtained was $$p = 3! \times 4/52 \times 4/51 \times 4/50$$ The first 3! is for the order of king, queen, jack. $4/52$ is the probability of drawing a king, $4/51$ is the probability…
Bootstrap
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stopped filtration = filtration generated by stopped process?

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky: Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration that it generates (unaugmented). Let $T$ be a bounded…
hoirkman
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Tighter tail bounds for subgaussian random variables

Let $X$ be a random variable on $\mathbb{R}$ satisfying $\mathbb{E}\left[e^{tX}\right] \leq e^{t^2/2}$ for all $t \in \mathbb{R}$. What is the best explicit upper bound we can give on $\mathbb{P}[X \geq x]$ for $x > 0$? A well-known upper bound…
Thomas
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Concerning an infinite server queue with Poisson arrivals

Here's the statement of the problem (from Ross's Introduction to Probability Models): For those unfamiliar with "infinite server queues," they are described here. In this case, however, the service times are not exponentially distributed; rather,…
thisisourconcerndude
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An extension of the birthday problem

Th birthday problem (or paradox) has been done in many way, with around a dozen thread only on math.stackexchange. The way it is expressed is usually the following: "Let us take $n$ people "independently" (no twins, etc.). What is the probability…
D. Thomine
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Is it unlikely to get the same number of heads/tails?

A question in probability by a non-mathematician: A fair coin is tossed $2N$ times. Is it unlikely that we get exactly $N$ heads and $N$ tails? From one side, this must be the most likely result! But intuitively, if someone reports to me that they…
Tom Y
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Hillary Clinton's Iowa Caucus Coin Toss Wins and Bayesian Inference

In yesterday's Iowa Caucus, Hillary Clinton beat Bernie Sanders in six out of six tied counties by a coin-toss*. I believe we would have heard the uproar about it by now if this was somehow rigged in her favor, but I wanted to calculate the odds of…
D. W.
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Can you create non transitive dice for any finite graph?

Let's say you have a finite directed graph, with no two nodes that point at each other. Can we assign each node a dice, so that each node beats the node it is pointing at. This is easy for acyclic graphs, but it is possible for some non-acyclic…
PyRulez
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Interesting shapes using probability and discrete view of a problem

Suppose we have a circle of radius $r$, we show the distance between a point and the center of the circle by $d$. We then choose each point inside the circle with probability $\frac{d}{r}$ , and turn it black (note that $\frac{d}{r}<1$). With these…
CODE
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Why does probability need the Axiom of pairwise disjoint events?

I'm a beginning student of Probability and Statistics and I've been reading the book Elementary Probability for Applications by Rick Durret. In this book, he outlines the 4 Axioms of Probability. For any event $A$, $0 \leq P (A) \leq 1$. If $\Omega…
Brother John
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Asymptotics for the expected length of the longest streak of heads.

As Introduction to Algorithms (CLRS) describes, the problem is Suppose you flip a fair coin $n$ times. What is the longest streak of consecutive heads that you expect to see? The book claims that the expects is $\Theta(\log{}n)$, and proves that…
Yai0Phah
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Most Probable Sum

Possible Duplicate: Probability of dice sum just greater than 100 A fair dice is rolled and the outcome of the face is summed up each time. We stop rolling when the sum becomes greater than 100. Which of the following is most probable…
Amol Sharma
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Distribution of weighted sum of Bernoulli RVs

Let $x_1,...,x_m$ be drawn from independent Bernoulli distributions with parameters $p_1,...,p_m$. I'm interested in distribution of $t=\sum_i a_ix_i,~a_i\in \mathbb{R}$ $m$ is not large so I can not use central limit theorems. I have the…
Alt
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Correlation Coefficient as Cosine

I've read that the correlation coefficient between two random variables may be viewed as the cosine as the angle between them, but I can't find any solid explanation. To be concrete, let $X$ and $Y$ be random variables on $(\Omega, \mathcal{F}, P)$…
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