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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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how to prove that ${S_N\over E[S_N]}$ converges to an exponential distribution

Suppose that $\{X_1,X_2,\ldots\}$ is a sequence of iid $L^1$-random variables such that $E[X_1]\neq 0$. Define for every $n$, $$ S_n=X_1+\cdots+X_n. $$ Let $N$ be a geometric random variable such that $$ P(N=k) = q^{k-1}p,\quad…
Ulton Prinsphield
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In probability, how can a sigma-algebra represent the total information?

Why does a sigma-algebra represent the information available at a given time? I understand the idea of filtration and stopping-time, given that each sigma-algebra represent the info we have at a specific time, but why is that? For instance in a game…
lezebulon
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What's the probability that Erica has one boy and one girl?

During a flight on an airplane, Eric strikes up a chat with Erica, the person sitting next to him. It turns out that Erica has two kids, and at least one of them is a girl born on a Tuesday. Being a mathematician, Eric decides to find the…
user19405892
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Expected Value Diverges?

Given the probability density function $f(x) = \frac{1}{x^2}$ for $x > 1$. The expected value of this function is: $$E[X] = \int_1^\infty \frac{1}{x} dx = \infty$$ Can someone please explain this? Is $E[X]$ only finite under certain condition? Is…
Paichu
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Generalizing the total probability of simultaneous occurrences for independent events

I want to generalize a formula and I need your help with this. This is not my homework or assignment but I need to come up with a concise formula that fits my documentation. Background for my problem: Considering all events to be independent of each…
TheCuriousEngineer
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A seeming paradox in a coin-flipping game

This is related to my other question on a similar topic. Suppose we play the following game: we flip a coin repeatedly and record the outcomes. For example we might get HHTTTHTTHHTTT.... Now Alice and Bob each choose distinct patterns of the same…
Fixee
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Probability of a boring afternoon

I am extremely bad in probabilities (as in so many other areas) and I fully understand that this question is off-topic and/or missing context (may I underline that this is not homework). Could you tell me how to compute the probability that, in a…
Claude Leibovici
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Rate of convergence of random series

$X_1,X_2,\cdots$ are iid with $E(|X_i|^{p})<\infty$ for some real $p\ge 1$ and $E(X_i)=\mu$. I am trying to find the largest $\alpha>0$ such that $n^{\alpha}\left[\dfrac{S_n}{n}-\mu\right]\to 0$ almost surely. I am able to find $\alpha$ for $p$…
Ergodic
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A and B play a series of games. Find the probability that a total of 4 games are played.

A and B play a series of games. Each game is independently won by A with probability $p$ and by B with probability $1 - p$. They stop when the total number of wins of one of the players is two greater than that of the other player. The player with…
idknuttin
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What is the probability that the letter came from LONDON?

A letter has come from exclusively LONDON or CLIFTON, but on the postmark only $2$ consecutive letters ''ON'' are found to be visible. What is the probability that the letter came from LONDON? This is a question of conditional probability. Let $A$…
learner_avid
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First to the sequence HT between two players

Two players, A and B, alternatively toss a fair coin (A tosses first and then B). The sequence of heads and tails is recorded. If there is a head followed by a tail (HT subsequence), the game ends and the person who tosses the tail wins. What is the…
Jojo
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Prove if $P(A | B^c) = P(A | B)$ then the events A and B are independent.

So I've started by saying that since $P(A | B^c) = P(A | B)$ we know that $\frac{P(A \cap B^c)}{P(B^c)} = \frac{P(A \cap B)}{P(B)}$. However I'm not sure where to go from there. Any help would be great!
mconn7
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On the Total Number of Tries Required to have $n$ Successes

The Problem A bag contains $b$ black balls and $w$ white balls. Balls are drawn at random from the bag until the last white ball is drawn. What is the expected number of balls drawn? My Partial Solution Suppose the balls are lined up in a line and…
fleur
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Expected norm of a random Gaussian vector

Let $X$ be a random vector in $\mathbb{R}^n$ whose entries are joint Gaussian with zero mean and covariance matrix $K.$ Is there a closed form expression for $\mathbb{E}||X||_2,$ as there is for the absolute deviation of a standard Gaussian in a…
Hedonist
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Calculating the probability to win with martingale in roulette

I've made a python program that uses martingale betting method to see what the probability is to double your starting pot and then stop at European roulette (no double 00). The program works like this, You got $100$ dollars in the beginning and the…
Thobias Ivarsson
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