# Tagged Questions

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### Distinct elements in the Union and Intersection of A and B

Take a set $x$ with $10$ distinct elements. Rule: Everytime you have two subsets, $A$ and $B,$ you also have $A\cup B$ and $A \cap B.$ What is the maximum number of subsets you can have such ...
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### How to prove the binomial theorem

I'm trying to prove the following: $$(p + q)^n = \sum_{x=1}^n \frac{n!}{x!(n-x)!}p^xq^{n-x}.$$ But I'm not sure where to start, would expanding the left hand side get me anywhere? Any tips or ...
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### Chance of playing a game

You are offered a chance to play a game. the rules are simple. there are $100$ cards face down. Of these, $55$ say win and $45$ say lose. You begin with $10000$ dollars. You must bet $1/2$ of your ...
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### Set Theory Proof [closed]

How can I prove that $A \cup B = A \cup (A'B)$ It is a useful identity when proving : $P(A \cup B) = P(A)+P(B)-P(AB)$ But I donĀ“t know where that identity comes from. Is there an analytical proof ...
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### If $X' \leq X$ almost surely, is it possible to prove that $P(X = s) \geq P(X' = s)$?

With respect to my previous question, let us define $X$ as: $$X = \sum_j^r l^j Y^j,$$ where $l^j \geq 0$ and $Y^j$, $j = 1, \ldots, r$ is a Bernoulli random variable which takes on values in ...
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### Mean and Standard Deviation self thought problem

I am 13 years old trying to teach myself about standard deviation and was wondering how this problem would look like. I know I am young to be learning this but I was reading about this and got ...
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### How can one pass from “almost surely” to “surely”?

Several results (e.g in probability theory or using prob. theory) are stated in an almost surely phrasing (meaning the set of outcomes where this is not so has measure zero) How can one pass from ...
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### How to prove something at Normal distribution

$X\sim N(\mu,\sigma^2)$. $A,B$ are constants and $A\ne0$. How to prove that $AX+B\sim N(A\mu+B,A^2\sigma^2)$ ? Thank you!
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### Question about $E(|Z|)$ at Normal distribution

$Z$ is a standard normal variable. How do I calculate $E(|Z|)$? ($E(Z)=0$). Thank you!
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### Probability of winning a head on a coin

The problem I am asking is generated from this problem: Carla and Dave each toss a coin twice. The one who tosses the greater number of heads wins a prize. Suppose that Dave has a fair coin ...