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Questions tagged [probability]

For questions about probability. independence, total probability and conditional probability. For questions about the theoretical footing of probability use [tag:probability-theory]. For questions about specific probability distributions, use [tag:probability-distributions].

The probability that an event occurs is a number in the interval $[0, 1]$, which represents how likely the event is to happen. $0$ indicates it will never happen, $1$ indicates it will always happen.

For example, throwing two dice gives a total of $6$ five times out of thirty-six. We write $$P(X=6)=\frac{5}{36}$$.

Use this tag for basic questions about probability, independence, total probability and conditional probability.

For questions about the theory of probability, use instead. For questions about specific probability distributions, use .

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Broken stick probability problem

We have all heard the old problem about forming a triangle from breaking a stick into three pieces, with the breaks randomly distributed. Some variations make the second break contingent on the first in some way. I present a new variation (not…
Potato
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Number of couples sitting at same table

The Problem $ab$ couples are sitting at $a$ tables with $2b$ seats each. Call a couple a "good" couple if they are seated at the same table. (1) What is the probability that, in total, there are exactly $k$ "good" couples? My Approach so…
Vincent Tjeng
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Probability of questions being on an exam

My girlfriend has an exam in her international development class tomorrow. She's been given $60$ terms to study (each takes a long time to learn thoroughly). Of those $60$ terms, $10$ will be on the exam, and she must discuss $3$ of them. Now,…
connorbode
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Why can 2 uncorrelated random variables be dependent?

I recently learned that two independent random variables $X$ and $Y$ must have a covariance of $0$. That means that the correlation between them is also $0$. However, apparently, the converse is not true. 2 random variables $X$ and $Y$ can have a…
David Faux
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Probability: Are disjoint events independent?

I just read that disjoint events, A, B, if, $\mathbb{P}(AB) = 0$ are independent. This really frustrates me. My teacher stated otherwise - $\mathbb{P}(AB) = 0 \iff A \cap B = \emptyset \implies \mathbb{P}(AB) = 0 \ne \mathbb{P}(A)\mathbb{P}(B)$…
Ivan Prodanov
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Expected area of triangle formed by three random points inside unit circle

Motivated by the discussion in The expected area of a triangle formed by three points randomly chosen from the unit square I tried to find an expression for the expected area of a triangle formed by three randomly chosen points inside the unit…
user164118
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Prove complements of independent events are independent.

Given a finite set of events $\{A_i\}$ which are mutually independent, i.e., for every subset $\{A_n\}$, $$\mathrm{P}\left(\bigcap_{i=1}^n A_i\right)=\prod_{i=1}^n \mathrm{P}(A_i).$$ show that the set $\{A_i^c\}$, that is the set of complements of…
Doria
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Expected number of tosses for two coins to achieve the same outcome for five consecutive flips

Consider two unbiased coins. Toss both until last 5 sequence outcome are same. That means we stop when output of the sequence of both are as follows: HTTHTHHTH , HHTTTHHTH. What is the expected number of trials?
user12290
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Doubling Money Game

The casino offers a certain win-lose game, where you have $p$ chance of winning. You can bet any amount of money, and if you win you get twice your bet; otherwise, you lose your bet. If you use the optimal strategy, what is your chance of doubling…
gambel
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Expected number of unique items when drawing with replacement

Having a set of size M, I'm drawing M items with replacement. What is expected number of unique items that got picked? Thanks, Jarek
Jarek Odzga
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Sums of two probability density functions

If the weighted sum of 2 probability density functions is also a probability density function, then what is the relationship between the random variables of these 3 probability density functions.
Xara
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Expectation on 1/X

In general can one say that for a random variable X: $E[\frac{1}{X}] = \frac{1}{E[X]}$ ? I've worked out a few examples where this works but I'm not sure how widely this is useful...
Dirk Calloway
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Why is $0.63212$ the probability of a $\frac1n$-probability event happening in $n$ trials?

I've always assumed on faulty intuition that if you have an event which occurs 1 in n chances, it will be super likely to happen at some point of that event occuring n times. However, given some analysis, it doesn't actually seem to be all that…
intuited
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$52$ cards reciprocal sum probability

Imagine a deck of $52$ cards but instead of having suits and ranks, they only have sequential (unique) integer ranks from $1$ to $52$. You could also imagine a standard deck of $52$ cards but convert the ranks and suits to an integer number from $1$…
David
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Probability of picking a random natural number

I randomly pick a natural number n. Assuming that I would have picked each number with the same probability, what was the probability for me to pick n before I did it?
Thomas
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