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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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The Tuesday Birthday Problem - why does the probability change when the father specifies the birthday of a son?

I've most recently read about the Tuesday Boy Problem via twitter and I, as probably most other people, was sure that the probability has to be 1/2. After having read through a lot of solutions which were not identical at all, I've come to the…
Huy
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Can conditional distributions determine the joint distribution?

Can conditional distributions determine the joint distribution? For example, let $X_1, \dots, X_n$ be random variables. Can their joint distribution be determined from the conditional distribution of $X_i$ given others, $i=1, \dots, n$? Can their…
Tim
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Prove ranks are uniformly distributed

We have n IID random variables $X_1, X_2, \ldots, X_n$. Let $R_i$ be $X_i$'s rank in the set $\{X_1, X_2, \ldots, X_3 \}$ when we order from large to small. How to prove $R_i, \forall i \in \{1, 2, \ldots, n\}$, is uniformly distributed on $\{1,…
NonalcoholicBeer
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If, tossing a coin 400 times, we count the heads, what is the probability that the number of heads is [160,190]?

I wanted to solve the problem with the Central Limit Theorem. Analyzing the question, I modeled the situation with a random variable : $$\begin{cases} 1 & \text{with probability } 1/2; \\ 0 & \text{with probability } 1/2; \end{cases} $$ Calculating…
BlacK
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A family has two children. One child is a girl. What is the probability that the other child is a boy?

My initial thought process: Sample space: GG, GB, BG, BB. I then crossed out BG because it's the same as GB because order doesn't matter here. And because we know one is a girl, there leaves two possibilities left. If the other is a boy, the…
iamashortgiraffe
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a part of expected value of Poisson distribution $E(X^2)=λ^2+λ$ proof?

a part of expected value of Poisson distribution : $E(X^2)=λ^2+λ$ What is the proof? (except using the Moment-generating function )
user106557
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Probability for pairing up

A set of 200 people, consisting of 100 men and 100 women, is randomly divided into 100 pairs of 2 each. Give an upper bound to the probability that at most 30 of these pairs will consist of a man and a woman. I intend to use the Chebyshev Inequality…
BVFanZ
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Does any moment generating function implies an existence of moments?

For a real-valued r.v. $\xi$ we suppose and existence of its moment generating function $m(t) = \mathsf E\mathrm e^{t\xi}$ for all $t\in (-h,h)$ where $h>0$. I wonder how many moments $M_n = \mathsf E\xi^n$ are finite. I know that using the…
SBF
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Probability of second ball being black

I was taking Caltech - ML Course and solving problem 1.3 in this link We have 2 opaque bags, each containing 2 balls. One bag has 2 black balls and the other has a black ball and a white ball. You pick a bag at random and then pick one of the balls…
Sachin Jain
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Calculating probabilities over longer period of time

There's a great question/answer at: Calculating probabilities over different time intervals This is an awesome answer, but I'd like to ask a related question: What if the period goes the other direction, for example, the probability is determined…
Neil Ticktin
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Tossing a coin until you have more heads than tails

I toss a coin a many times, each time noting down the result of the toss. If at any time I have tossed more heads than tails I stop. I.e. if I get heads on the first toss I stop. Or if toss T-T-H-H-T-H-H I will stop. If I decide to only toss the…
James
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Alice and Bob sometimes lie; a die is thrown and they claim different results; what is the probability that Bob is being honest?

Alice speaks the truth with probability $3/4$ and Bob speaks the truth with probability $2/3$. A die is thrown and both Alice and Bob observe the number. Afterwards, Alice asserts to Carl (who does not know the number) that the number is $3$ while…
MangoPizza
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Probability of spelling IDAHO correctly

I'm working with a problem stated as follows: A sign reads IDAHO. Two letters are removed and put back at random, each equally likely to be put upside down as in the correct orientation. What is the probability that the sign still reads IDAHO? So,…
Tanamas
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Techniques for determining how "random" a sample is?

What techniques exist to determine the "randomness" of a sample? For instance, say I have data from a series of $1200$ six-sided dice rolls. If the results were 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, ... Or: 1, 1, 1, ..., 2, 2, 2, ..., 3, 3, 3,…
Scott Mitchell
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Throwing All Numbers From 2 to 12 With Two Dice

What is the expected number of times one must throw two fair dice before all numbers from 2 to 12 have appeared at least once?
ThudanBlunder
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