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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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A certain country has four regions: North, East, South, and West. ...

A certain country has four regions: North, East, South, and West. The population of these regions are 3 million, 4 million, 5 million, and 8 million, respectively. There are 4 cities in the North, 3 in the East, 2 in the South, and there is only 1…
user102156
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Probability that 3 darts land in a same half of a dart board

3 darts are thrown (equal probability of landing anywhere on the dart board). What's the probability that they all land on a same half of the dart board? Edit: I know the first 2 darts can land anywhere but don't know how to find the probability…
lollab
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Why is $P(a \text{ and } b)$ maximized when $P(a \text{ or } b)$ is minimized?

I can't seem to wrap my head around why $P(a \text{ and } b)$ is minimized when $P(a \text{ or } b)$ is maximized. This comes from PIE: $$P(a \text{ or } b) = P(A) + P(B) - P(a \text{ and } b).$$ Can someone please explain the intuition behind this?…
Eric Aldrin
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Probability of an even number

I have been having problems solving the following problem. Two real numbers $x$ and $y$ are chosen uniformly at random in the interval $(0,1)$. What is the probability that the closest integer to $x/y$ is even? I wrote a computer program to…
user66151
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If I reset after each run, how many times do I need to repeat an experiment to have a 75% chance of running it on all the subjects?

Suppose there is a bag with 100 marbles. And I can draw 5 marbles in one attempt. But after that draw I have to put the marbles back. In how many attempts do I have a 75% chance that I have drawn each marble at least once? I wrote a brute force…
Wes
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Records and covers

Juan is challenging his friend Thomas with the following: Juan has 5 LP vinyl records, each of them of a different band, and asks Thomas to match them with its respective band name, by showing him only the cover photo, not the names etc. Thomas…
Martine Stephant
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rolling dice - win if the sum of rolls is exactly $n$

This question was asked during my interview: Suppose you have a fair dice (6 faces as usual). You can pick a positive integer $n$. Then you can repeatedly roll a dice until the sum of the rolls exceeds or equals to $n$. If the sum is exactly $n$, it…
Ted
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If two random customers pick four chopsticks at random, what is the probability that they pick one for each color?

Did I solve this probability problem correctly? The question goes like this: In every table in Earl’s Diner, there are exactly four pairs of chopsticks, and each pair is uniquely colored. If two random customers pick four chopsticks at random, what…
Jill and Jill
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Average time for a tetris field to fill

I have a tetris field, infinitely wide (irrelevant) and N blocks high. Now random blocks start falling down, and I do not move them. Given that the field is N blocks high, the blocks aren't moved, the blocks fall one block per second and the blocks…
orlp
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How do factor graph and sum-product algorithm work?

I'm reading a tutorial on factor graph and its sum-product algorithm. The tutorial is at http://www.isiweb.ee.ethz.ch/papers/arch/aloe-2004-spmagffg.pdf. What I don't understand is the example on page 19. I don't know how they come up with these…
v4r
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Conditional probabilities from a joint density function

The joint density function of two continuos random variables $X$ and $Y$ is given by: $f(x,y) = 8xy$ if $0\le y\le x\le 1$ and $0$ otherwise. Calculate $P(X \le \frac{1}{2})$ Calculate $P(Y \le \frac{1}{4} \mid X = \frac{1}{2})$ Calculate the…
dreamer
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Probability question about chords on a circle

Four points are chosen independently and at random on a circle. Find the probability that chords X1X2 and X3X4 intersect a) without calculation using a symmetry argument and b) from the definition by an integral I'm lost here. I'm thinking of using…
user9752
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Interpreting the word "randomly"

Suppose that 3 indistinguishable balls are placed at random into 3 distinguishable cells. What is the probability that exactly one cell remains empty? The book's answer is $$\frac{3(3-1)}{3+3-1\choose{3}}=\frac{3}{5}$$. In words, the number of ways…
johnson
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How much should I pay for a chance to win 100$?

There are $4$ closed doors, with $100\$$ behind one of them. You can pay $X$ to open a door. If the money is there, you can keep it. If not you can pay another $X$ to open the next door, and so on. What is the most I can pay and still win on…
badmax
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What is the tail $\sigma$-field?

In the book "probability : example and application", they define a tail $\sigma$-field as $\mathcal T=\bigcap_{n=1}^\infty \mathcal F_n'$ where $\mathcal F_n'=\sigma (X_n,X_{n+1},...)$. They call $\mathcal t$ remote future and say that $A\in…
user659895
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