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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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martingale and expectation

The following is an old exam problem: Let $\{X_n\}$, $n\geq0$, be a process adapted to a filtration $F_n$. Prove that $(X_n,F_n)$ is a martingale, if and only if for all bounded $F_n$-stopping time $\tau$, $EX_{\tau}=EX_0$ holds. I know if $X_n$ is…
guest625
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Probability of land ownership

During a research related to economy of land ownership, I ran into an interesting probability problem: There are N citizens and N land-plots. A. Initially, each land-plot is given to a citizen selected at random. B. Then, each land is sold with…
Erel Segal-Halevi
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Why binomial distribution doesn't count permutations?

Why in Binomial distribution the formula starts with $n\choose k$ and not with something like $k!\over n!$? Isn't the order important? Or, it is important but due to the independence of the event?
gbox
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How biased is this biased coin

Suppose that we have a coin that we suspect is biased, but that we don't know precisely how biased it is: all we know is that its probability p of landing heads is some fixed value between .4 and .6, inclusive. We flip the coin 100 times, and it…
Dave
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Expected number of collisions

$n$ items are distributed into $n$ boxes such that each item is independently put with probability $p_j$ to be put in box $j$, for $j=1,2,\ldots,n$, where $\sum_{j=1}^np_j=1$. A collision occurs whenever an item is put into a non-empty box.…
strake
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Convergence of a sum of random variables

Let $(X_n)$ be a sum of i.i.d. positive random variables such that $\mathbb{E}(X_1)=1$ and $\mathbb{P}(X_1\neq 1)>0$. Put $M_n=X_1\ldots X_n$. Show that $\sum _{n\geq 1}\sqrt{M_n}< +\infty $ a.e. It can be show that $M_n$ is a martingale so that…
Patissot
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Probability when cutting the stick twice

Given a stick of length $l$. We cut the stick twice. Let $X$ be the random variable defined by the length of the stick after the first cut, and $Y$ be the random variable defined by the length of the stick after the second cut. What is the…
Tien Kha Pham
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what is the probability that the circumcircle of 3 point

Mary picks any three non-collinear points inside a given circle, what is the probability that the circumcircle of these 3 points will be covered by the original circle? This is from a test question a few months ago.I got the result is…
user223800
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probability that the $100^{th}$ ball you pick is black given the details inside

We begin with a bag containing 3 white balls and another 5 black balls. After each ball is picked, it is returned AND you add to the bag another 4 balls of the same color. For example, the probability that the second ball I pick is black…
johni
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Average distance between 2 points on surface of sphere?

How can I find an average distance between two points lying on surface of a sphere of a certain radius? More importantly : can knowing the average distance between two points on surface of a disk ( this question has already an answer on MSE) be…
jimjim
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Roll eleven dice such that the product is prime

So the problem is: What is the probability of rolling eleven dice such that their product is prime. The dice is numbered from 1 to 6 and there is an equal chance of getting each number. So in order for the product to be prime, all but one of the…
Gates VVang
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What's the probability that there's at least one ball in every bin if 2n balls are placed into n bins?

I've been working on this all day long. Here's what I've done until now.The denominator is easy. It's $n^{2n}$. I compute the numerator as follows. All $n$ bins have at least one ball = $n$ bins must have one of the $2n$ balls each + the remaining…
Ragavan N
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A fun card game involving probability, getting all 13 ranks (any suit(s)) vs. 5 in a row of red or black.

Two people, (call them C and D), decide to play a card game for fun. They use an ordinary fair deck of $52$ cards, shuffled well before each hand is drawn, and randomly draw cards from it one a time without replacement, both using (sharing) the…
David
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Probability of throwing balls into bins.

Suppose we have $M$ balls placed randomly into $N$ boxes, wherein each ball has an equal chance of landing in each bin. How would we go about finding the expected number of balls in the first box? I assumed we could use a binomial distribution,…
FinneganJ
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Probability question on finding a defective ball in a specific box

There are two boxes, each containing two balls. Each ball is defective with probability 1/4, independent of other balls. The probability that exactly one box contains exactly one defective ball is (A) 3/8 (B) 5/8 (C) 15/32 (D) 17/32 One box can…
Day Dreamer
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