Questions tagged [probability]

For basic questions about probability and the questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities. For questions about the theoretical footing of probability (especially using [tag:measure-theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

Probability is a numerical quantity which lies in the interval $[0, 1]$. According to the Bayesian view, it represents the belief about an event or preposition and it is interpreted as how likely it is for an event to occur, or of how likely it is for a proposition to be true. According to the frequentist view, it represents the relative frequency of a favourable event with respect to the sample space. Use this tag for basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantities. For questions about the theoretical footing of probability (especially using ), please ask under instead. For questions about specific probability distributions, please use .

95623 questions
72
votes
9 answers

What do $\pi$ and $e$ stand for in the normal distribution formula?

I'm a beginner in mathematics and there is one thing that I've been wondering about recently. The formula for the normal distribution is: $$f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\displaystyle{\frac{(x-\mu)^2}{2\sigma^2}}},$$ However, what are $e$…
pimvdb
  • 1,213
  • 2
  • 12
  • 15
71
votes
3 answers

How likely is it not to be anyone's best friend?

A teenage acquaintance of mine lamented: Every one of my friends is better friends with somebody else. Thanks to my knowledge of mathematics I could inform her that she's not alone and $e^{-1}\approx 37\%$ of all people could be expected to be in…
71
votes
2 answers

Numerical phenomenon. Who can explain?

I was doing some software engineering and wanted to have a thread do something in the background to basically just waste CPU time for a certain test. While I could have done something really boring like for(i < 10000000) { j = 2 * i }, I ended up…
Jake Mirra
  • 2,699
  • 11
  • 22
70
votes
10 answers

There are 4 cups of liquid. Three are water and one is poison. If you were to drink 3 of the 4 cups, what is the probability of being poisoned?

In Season 5 Episode 16 of Agents of Shield, one of the characters decides to prove she can't die by pouring three glasses of water and one of poison; she then randomly drinks three of the four cups. I was wondering how to compute the probability of…
Eli
  • 932
  • 1
  • 7
  • 9
70
votes
6 answers

Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?

In the book "Zero: The Biography of a Dangerous Idea", author Charles Seife claims that a dart thrown at the real number line would never hit a rational number. He doesn't say that it's only "unlikely" or that the probability approaches zero or…
regularmike
  • 715
  • 5
  • 8
69
votes
6 answers

Chance of meeting in a bar

Two people have to spend exactly 15 consecutive minutes in a bar on a given day, between 12:00 and 13:00. Assuming uniform arrival times, what is the probability they will meet? I am mainly interested to see how people would model this formally. I…
Beltrame
  • 2,976
  • 1
  • 25
  • 33
68
votes
5 answers

How can a probability density be greater than one and integrate to one

Wikipedia says: The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one. and it also says. Unlike a probability, a probability density function can take on values greater than one; for…
zenna
  • 1,389
  • 1
  • 11
  • 15
66
votes
1 answer

Distinguishing probability measure, function and distribution

I have a bit trouble distinguishing the following concepts: probability measure probability function (with special cases probability mass function and probability density function) probability distribution Are some of these interchangeable? Which…
Marc
  • 3,057
  • 2
  • 17
  • 27
65
votes
7 answers

Poisson Distribution of sum of two random independent variables $X$, $Y$

$X \sim \mathcal{P}( \lambda) $ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim \mathcal{P}( \lambda + \mu)$ but I don't understand how to…
user31280
65
votes
6 answers

If we randomly select 25 integers between 1 and 100, how many consecutive integers should we expect?

Question: Suppose we have one hundred seats, numbered 1 through 100. We randomly select 25 of these seats. What is the expected number of selected pairs of seats that are consecutive? (To clarify: we would count two consecutive selected seats as a…
David Steinberg
  • 1,340
  • 11
  • 22
65
votes
4 answers

Product of two Gaussian PDFs is a Gaussian PDF, but Product of two Gaussian Variables is not Gaussian

The Product of Two Gaussian Random Variables is not Gaussian distributed: Is the product of two Gaussian random variables also a Gaussian? Also Wolfram Mathworld So this is saying $X \sim N(\mu_1, \sigma_1^2)$, $Y \sim N(\mu_2, \sigma_2^2)$ then…
Frames Catherine White
  • 1,390
  • 1
  • 11
  • 23
64
votes
5 answers

Probability for the length of the longest run in $n$ Bernoulli trials

Suppose a biased coin (probability of head being $p$) was flipped $n$ times. I would like to find the probability that the length of the longest run of heads, say $\ell_n$, exceeds a given number $m$, i.e. $\mathbb{P}(\ell_n > m)$. It suffices to…
Sasha
  • 68,169
  • 6
  • 133
  • 210
64
votes
12 answers

If a coin toss is observed to come up as heads many times, does that affect the probability of the next toss?

A two-sided coin has just been minted with two different sides (heads and tails). It has never been flipped before. Basic understanding of probability suggests that the probability of flipping heads is .5 and tails is .5. Unexpectedly, you flip the…
skurwa
  • 777
  • 1
  • 5
  • 8
64
votes
17 answers

Is the Law of Large Numbers empirically proven?

Does this reflect the real world and what is the empirical evidence behind this? Layman here so please avoid abstract math in your response. The Law of Large Numbers states that the average of the results from multiple trials will tend to converge…
62
votes
8 answers

Probability that random moves in the game 2048 will win

I have recently played the game 2048, created by Gabriele Cirulli, which is fun. I suggest trying if you have not. But my brother posed this question to me about the game: If he were to write a script that made random moves in the game 2048, what…
N. Owad
  • 6,141
  • 3
  • 19
  • 40