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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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question about conditional probability for continuous r.v

This is related to the content in the book by Grimmett and Stirzaker "Probability and random processes" 3rd ed. On page 111, it calculated, as an example, the conditional density function of $X_1+X_2$ given $X_1=X_2$ for two i.i.d. exponential r.v.…
Qiang Li
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$\mathbb{P}(B) = 1 \implies \mathbb{P}(A \mid B) = \mathbb{P}(A)$

Suppose I have two events $A$, $B$ from the same sample space with $\mathbb{P}(B) = 1$ and $\mathbb{P}(A) > 0$. How can I show that $$\mathbb{P}(B) = 1 \implies \mathbb{P}(A \mid B) = \mathbb{P}(A)\text{?}$$ The definition of conditional probability…
Clarinetist
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Coin flipping probability taking turns between players

I've worked out the result to a basic coin flipping problem and want to generalize it. The basic problem is: there are N players, and they take turns of flipping a coin (in the same cyclical order) until the first person to get 1 heads wins. Coin…
sambajetson
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How to apply Chernoff's bound when variables are not independent

Let $X = \sum_{i=1}^n{X_i}$, for Bernoulli random variables $X_i$ which are not necessarily independent. However, assume that conditioned on any possible values for the other variables, the probability that $X_i = 1$ is at most $p$. I would like to…
Zur Luria
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A probability question that uses the binomial expansion

The question is as follow: (i) Find the binomial expansion of $(1-x)^{-3}$ up to and including $x^{4}$. (ii) A player throws a 6-sided fair die at random. If he gets an even number, he loses the game and the game ends. If he gets a "1", "3" or "5"…
user42268
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100 pieces of paper in a box, one of which has a black dot on it. Probability Question.

There are $100$ pieces of paper in a box, one of which has a black dot on it. If $100$ people go up one by one and pick a paper from the box, which one has the lowest probability of getting the black dot, and which one has the highest probability of…
Tekheny Ghemor
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Is $\{\frac1n\sum_{k=1}^n X_k\ \text{converges}\}$ a tail event?

Suppose that $X_1,X_2,\dots$ is a sequence of random variables on some probability space. The tail $\sigma$-algebra $\mathcal{T}$ is defined as the intersection of $\sigma$-algebras $\mathcal{T}:=\bigcap_n\mathcal{F}_n$, where…
Xiang Yu
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Difficult probability of choosing ball from bag with $7$ balls labelled from $1-7$

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with probability, which perhaps yields the shortest, simplest proofs, but other than that, the textbook gave no hints really and…
anonymous
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Calculating the probability of an event giving the union and complement

There are two independent events. The probability that both occurs at the same time is $\frac{1}{6}$ and the probability that none of them happens is $\frac{2}{3}$. What is the probability that only one of them occurs? I'm trying to solve it but I…
Rods2292
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A question about probability

I have met a interesting question: If today rains, the probability that tomorrow rains is $0.6.$ If today doesn't rain, the probability that tomorrow rains is $0.2.$ Given Tuesday rained, what's the probability that Monday rained? I have no idea…
JSCB
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Conditional independence if and only factorization applies

Murphy, in his Machine Learning: A Probabilistic Perspective, defines two conditionally independent random variables essentially as follows: $X$ and $Y$ are conditionally independent given $Z$, i.e., $X \perp Y \mid Z$, if and only if $$p(X, Y \mid…
Clarinetist
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Find the probability of three tosses of a fair coin

Find the probability that, in three tosses of a fair coin, there are three heads, given that there is at least one head. I manage to get $\frac{3}{6}$ or $\frac{1}{6}$ but the right answer is $\frac{1}{7}$ I have no idea, Can you please…
Cin Sb Sangpi
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What happens when $X$ equals its own expected value?

Let $X$ denote a random variable with “moments” $M_1:=E(X)$, $M_2:=E(X^2)$, . . . (the first four of which, at least, are assumed to be finite). Show that $M_4+6M_2(M_1)^2\ge 4M_3(M_1)+3(M_1)^4$ Under what circumstances would you get equality? I…
JMA
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What is the probability of selecting five of the winning balls and one of the supplementary balls?

So I'm just doing a bit of probability questions and wanted to make sure I got it right. I have $50$ balls numbered $1-50$, and we pick $6$ winning balls and $2$ supplementary without replacement. So the chance to get the $6$ winning balls would…
Rivasa
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