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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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Summing over conditional probabilities

I've seen the following formula being used in various places: $P(a|b) = \sum_z P(a|z) \times P(z|b)$ So essentially they sum over all the variables z to get the original conditional probability. Is this correct, and does it make any strong…
Marek
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Discover to which batch a coin belongs

The following question is taken from a book, in a chapter on probabilty: You have two batches of unbalanced coins. One has coins which turn up head with probability $p_1$, and the other has coins that turn up head with probability $p_2$. All you…
SomeStrangeUser
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Minimum number of flips to guarantee heads

This is a weird problem that popped into my head: given a fair coin, how many flips is required to guarantee heads? If I get a tails, then another tails, and another etc., the chance of getting a heads increases every time. But there is still a…
imas145
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Why is the probability that a continuous random variable takes a specific value zero?

My understanding is that a random variable is actually a function $X: \Omega \to T$, where $\Omega$ is the sample space of some random experiment and $T$ is the set from which the possible values of the random variable are taken. Regarding the set…
davitenio
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Magic 8 Ball Problem

This problem is probably simple enough to have an analogous problem, I just don't know the name so I'm going to describe it and hopefully somebody can point me in the right direction. The problem is this: estimating the number of sides of a Magic 8…
aardvarkk
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Given two randomly chosen natural numbers, what is the probability that the second is greater than the first?

Suppose that there exists some apparatus that, when prompted, displays a random natural number (i.e. it picks an integer uniformly from the range $[1, \infty)$). A man writes down a number generated by this apparatus, denotes it as the first number,…
Peter Olson
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Probability distribution of the maximum of random variables

Given $N$ random iid variables, $X_1, \ldots, X_N$, with a uniform probability distribution on $[0, 1)$ what is the distribution of $\displaystyle \max_{i = 1 \ldots N}(X_i)$?
Alexandru
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Difference between Real Analysis and Probability Theory?

I do not really see a big difference between the two subjects. I was wondering if somebody can explain what the big difference between them is. Let us compare the superficial differences: In real analysis our subsets are called "measurable sets",…
Nicolas Bourbaki
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Difference between Probability and Probability Density

This question is from DeGroot's "Probability and Statistics" : Unbounded p.d.f.’s. Since a value of a p.d.f.(probability density function) is a probability density, rather than a probability, such a value can be larger than $1$. In fact, the…
Silent
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How to reasonably estimate the probability of your father being exactly 12222 days older than you?

On 2024/05/03, my brother is 10,000 days of age, while my father will be 22,222 days old. To tell them as a fun-fact, I would like to grab a feel of how special this is. I have no access to datasets that could provide me a distribution of the…
Pieter Depamelaere
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Probability of an event happening N or more times

I need to determine the probability of an event happening N or more times over M iterations. I know the probability of the event in question happening and its likelihood does not change over time. For example, say I am rolling a six-sided die and I…
Scott Mitchell
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Simulating a fair coin toss using a biased coin in fixed number of tosses

For which values of $p$ can we simulate a fair coin toss using a fixed number of tosses of a $p$-biased coin? Here are some positive and negative examples: When $p = 1/2$, this is trivially possible. When $p = 1/2 \pm 1/\sqrt{12}$, this is…
Yuval Filmus
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Probability Problem with $n$ keys

A woman has $n$ keys, one of which will open a door. a)If she tries the keys at random, discarding those that do not work, what is the probability that she will open the door on her $k^{\mathrm{th}}$ try? Attempt: On her first try, she will have the…
CAF
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Probability that rectangle is inside a circle

We randomly uniformly pick point $x$ on a unit circumference $\{x^2 + y^2 = 1\}$. Then we randomly uniformly pick point $y$ on a unit circle $\{x^2 + y^2 \leq 1\}$. Let $R$ be a rectangle with diagonal $xy$, which sides are parallel to coordinate…
Wunsch Punsch
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Why is intersection of two independent set probability a multiplication process?

Why is the probability of intersection of two independent sets $A$ and $B$, a multiplication of their respective probabilities i.e. Why is $$\mathbb{P}(A \cap B) = \mathbb{P}(A) \cdot \mathbb{P}(B)?$$ this question is about the intuition behind the…
user1675999
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