# 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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### Very quick question about random variables (Formal writing).

I've came across this confusion while reading this question: The time until Sam receives his first email is exponentially distributed with parameter $4$. The time between the first email and the ...
27 views

### Law of total probability proof

I am trying to prove $P(A_1\cap A_2\,\cap ...\cap A_n)=P(A_1)P(A_2|A_1)P(A_3|A_1\cap A_2)\,\cap ... \cap\,P(A_n|A_1\cap A_2\,\cap ...\cap A_{n-1})$ assuming that the conditional probabilities exist. ...
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### K consecutive heads in N tosses USING COMBINATRONICS

I would like to understand how to approach questions as: Find the Probability of observing K consecutive heads in N tosses, but I am interested only in the combinatorics/permutations approach, since I ...
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### Is $(y,T)=(7,8)$ the only solution in positive integers to $\frac{F(y)}{F(T)}=\frac12$, where $F(x)=x(x-1)(x-2)(x-3)$?

Define $F(x)=x\cdot(x-1)\cdot(x-2)\cdot(x-3)$. Let $y,T$ be positive integers. Is it then true that $y=7$, $T=8$ is the only solution to this equation? $$\frac{F(y)}{F(T)}=\frac{1}{2}$$ Motivation ...
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### Two sequences of measures $(\mu_n)_{n \geq 0}$ and $(\nu_n)_{n \geq 0}$ weakly converging to singular measures $\mu$ and $\nu$

I have two sequences of positive finite measures $(\mu_n)_{n \geq 0}$ and $(\nu_n)_{n \geq 0}$ that are absolutely continuous w.r.t. Lebesgue measure, more precisely \begin{equation*} \mu_n(dx) = a_n^...
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### 5 consecutive heads in 25 coin tosses with linearity of expectation

I am reasoning around the linearity of expectation. If for example, I want to know the expected number of a pair HH in 25 tosses (note that HHH has 2 possible HH pairs) I could use linearity of ...
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### Minimizing the variance in a variant of bagging Weighted Aggregation(Wagging)

In our machine learning course we have learned Bagging, wherein A variant of bagging call Weighted Aggregation is introduced, where the result is a weighted sum of all the estimators instead of ...
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### prove S as a probability distribtuion summation to 1 on a given variable [closed]

s(A, B|C, D) as: q(A, B|C, D) = Sum(p(A|C', B, D).p(C'|D).p(B|C,D)), Sum on C' in the above equation. show s(A, B|C, D) as valid probability distribution. Please show pointers on how to proceed. Thank ...
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### How do I calculate the probability of getting a flush from a 52 card deck using mathematica? [closed]

I know how to calculate this with pen and paper but would like to do the same in mathematica. However I'm new to mathematica so I have no idea where I'm supposed to start. Watched a couple of YouTube ...
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### Alternative solution to license plates question

I'm studying probability and I tried to think an alternative approach to solve this question. The statement is as follows: How many different 7-place license plates are possible when 3 of the entries ...
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### Joint likelihood of n samples iid from a binomial distribution vs joint probability, and the lack of a binomial coefficient

Let's assume I have 4 observations with each observation is modelled as a bernoulli trial with probability $p$. Sucesses are labelled as 1, failure is 0. My observations $(x_1, x_2, x_3, x_4)$ are as ...
### Closed-form formula for $u(t):= \mathbb E[h(X_1)h(tX_1+(1-t^2)^{1/2} X_2)]$, where $(X_1,X_2,\ldots,X_d)$ is uniform on sphere and $h(x):=(x+c)^k$
Fix $c \in \mathbb R$ and an integer $k \ge 0$, and consider the function $h:\mathbb R \to \mathbb R$ defined by $h(x) := (x+c)^k$. Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of ...