# 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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### Is it possible to have $\limsup_{n\to \infty}A_n = \emptyset$? [closed]

Given a sequence of events $(A_1, A_2, A_3,\ldots)$, is it possible that: $$\limsup_{n\to \infty} A_n = \{𝜔∈Ω:∀𝑛∈N,∃𝑘≥𝑛 | 𝑠.𝑡:𝜔 ∈𝐴_k\} = \emptyset$$ If it's possible kindly show me an example.
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### The Plausibility of Cavemen "Inevitably Discovering" Mathematical Principles

Recently, I had the following idea about the inevitableness of a prehistoric caveman "discovering" fundamental principles of mathematics based on the laws of physics and nature of the ...
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### Expected value and correlation

I have the following table, where the first row are the intervals of $y$ variable and the first column are the intervals of the $x$ variable. After getting 4000 samples, I placed their count into the ...
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### Are these notations about convergence in distribution standard?

I wanted to ask if the following notations make sense and/or are used. Convergence in distribution of a sequence $X_n$ of real random variables to the random variable $X$ is often indicated like this: ...
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### Probability that a determinant is equal to zero about 2*2 or 3*3 Integer matrix.

In fact, it is an expansion of this problem. But I restricted the elements of the matrix to be integers only. Obviously this probability is related to the range of random $\text{item}\in[0,n]$. For ...
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### Joint Probability Distribution, Find $c$

I am having a little bit of trouble with this practice question and I would like some insight of how to proceed. If $X$ and $Y$ are discrete random variables with joint probability: \begin{equation*} ...
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### Expected value of $E[X_1X_2]$ of a stochastic process

Say I have a transition matrix $P$ of a stochastic process and an initial distribution vector $\nu$. Is the expected value $E[X_1X_2]$? $$E[X_1X_2] = \sum x_1.x_2P(X_1=x_1 \cap X_2=x_2).$$
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### Properties of Expectations. [closed]

If X1, X2 are the results of rolling a fair, six-sided die twice. How do I show that E[min(X1, X2)] = 1/2 E[X1 + X2 − |X1 − X2|]. Also, how do I compute the value of this expectation E?
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### Is it possible to calculate the odds for a one more goal to be scored in the first half from the odds of the teams winning the first half?

Let's say that at 35 minutes of the first half in one Bookmaker we find the following odds for the first half result: ...
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### Approach of solving a probability question involving independent events not matching that given in my book.

Question: A and B are two independent events. The probability that both A and B occur is $\frac{1}{6}$ and the probability that at least one of them occurs is $\frac{2}{3}$​. The probability of the ...
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### Higher derivatives of the log-partition function

I need higher derivatives of the log-partition function $Z(z)=\log \sum_i \exp(z_i)$, has anyone derived the formula? Looking at concrete values of derivatives up to order 8, evaluated at $z=(1,1,1)$ ...
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### $|W_{t_n}| \to \infty$ a.s. as $n \to \infty$ for Brownian $(W_t)_{t \in \mathbb{R}_+}$ with $\sum_{n=1}^\infty t_n^{-0.5} < \infty$?

Let $(W_t)_{t \in \mathbb{R}_+}$ be a Brownian motion. Let $(t_n)_{n \in \mathbb{N}} \subset \mathbb{R}_+$ be a sequence of time points, such that $\sum_{n=1}^\infty t_n^{-0.5} < \infty$ (so, for ...
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### Chances of a "Straight" in a 13 Card Hand [closed]

If dealt $13$ a card hand from a standard $52$ card deck; what is the probability that the $13$ card hand contains a $5$ card straight.
Let $(X,A,\mu)$ be a set, a $\sigma$-algebra and a measure. Suppose that $\mu(X) = 1$. Let $u : X \rightarrow {\mathbb{R}}$ and $f : \mathbb{R} \rightarrow \mathbb{R} \, \cup \, \{+\infty \}$ be an ...
### simple walk on $\mathbb{Z}$
Consider the simple random walk $(X_n)_{n \in \mathbb{N}}$ starting from $X_0 = 0$. Consider $\varepsilon>0$, show that, for all $\delta>0$,  \lim _{n \rightarrow \infty} \mathbb{P}\left(\frac{...