A probability provides a quantitative description of the likely occurrence of a particular event.
65
votes
7answers
18k views
What is the difference between “likelihood” and “probability”?
The wikipedia page claims that likelihood and probability are distinct concepts.
In non-technical parlance, "likelihood" is usually a synonym for "probability," but in statistical usage there is a ...
49
votes
6answers
3k views
Why does a 95% CI not imply a 95% chance of containing the mean?
It seems that through various related questions here, there is consensus that the "95%" part of what we call a "95% confidence interval" refers to the fact that if we were to exactly replicate our ...
29
votes
16answers
6k views
What's the difference between probability and statistics?
What's the difference between probability and statistics, and why are they studied together?
29
votes
6answers
2k views
Motivation for Kolmogorov distance between distributions
There are many ways to measure how similar two probability distributions are. Among methods which are popular (in different circles) are:
the Kolmogorov distance: the sup-distance between the ...
25
votes
3answers
3k views
A Probability distribution value exceeding 1 is OK?
On the Wikipedia page about naive bayes classifiers here there is this line "P(height|male) = 1.5789 (A probability distribution over 1 is OK. It is the area under the bell curve that is equal to 1.)" ...
24
votes
12answers
1k views
The Monty Hall Problem - where does our intuition fail us?
From Wikipedia :
Suppose you're on a game show, and
you're given the choice of three
doors: Behind one door is a car;
behind the others, goats. You pick a
door, say No. 1, and the host, ...
24
votes
7answers
1k views
What is the probability that this person is female?
There is a person behind a curtain - I do not know whether the person is female or male.
I know the person has long hair, and that 90% of all people with long hair are female
I know the person has a ...
22
votes
12answers
2k views
What is the most surprising characterization of the Gaussian (normal) distribution?
A standardized Gaussian distribution on $\mathbb{R}$ can be defined by giving explicitly its density:
$$ \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$
or its characteristic function.
As recalled in this ...
20
votes
3answers
1k views
Why law of large numbers does not apply in the case of Apple share price?
Here is the article in NY times called "Apple confronts the law of large numbers". It tries to explain Apple share price rise using law of large numbers. What statistical (or mathematical) errors does ...
20
votes
8answers
4k views
What is Bayes' theorem all about?
What are the main ideas, that is, concepts related to Bayes' theorem?
I am not asking for any derivations of complex mathematical notation.
19
votes
14answers
2k views
Are there any good movies involving mathematics or probability?
Can you suggest some good movies which involve math, probabilities etc? One example is 21. I would also be interested in movies that involve algorithms (e.g. text decryption). In general "geeky" ...
19
votes
4answers
1k views
Probability of not drawing a word from a bag of letters in Scrabble
Suppose you had a bag with $n$ tiles, each with a letter on it. There are $n_A$ tiles with letter 'A', $n_B$ with 'B', and so on, and $n_*$ 'wildcard' tiles (we have $n = n_A + n_B + \ldots + n_Z + ...
19
votes
2answers
881 views
Does the distribution $\log(1 + x^{-2}) / 2\pi$ have a name?
I ran across this density the other day. Has someone given this a name?
$f(x) = \log(1 + x^{-2}) / 2\pi$
The density is infinite at the origin and it also has fat tails. I saw it used as a prior ...
18
votes
6answers
896 views
How likely am I to be descended from a particular person born in the year 1300?
In other words, based on the following, what is p?
In order to make this a math problem rather than anthropology or social science, and to simplify the problem, assume that mates are selected with ...
17
votes
5answers
3k views
Convergence in probability vs. almost sure convergence
I've never really grokked the difference between these two measures of convergence. (Or, in fact, any of the different types of convergence, but I mention these two in particular because of the Weak ...
17
votes
1answer
878 views
Differences between a statistical model and a probability model?
Applied probability is an important branch in probability, including computational probability. Since statistics is using probability theory to construct models to deal with data, as my understanding, ...
17
votes
3answers
404 views
Elementary statistics for jurors
I have been summoned for jury duty. I am conscious of the relevance of statistics to some jury trials. For example, the concept of "base rate" and its application to probability calculations is ...
16
votes
3answers
2k views
Amoeba Interview Question
I was asked this question during an interview for a trading position with a proprietary trading firm. I would very much like to know the answer to this question and the intuition behind it. Thank you! ...
16
votes
3answers
918 views
Logistic Regression and Dataset Structure
I am hoping that I can ask this question the correct way. I have access to play-by-play data, so it's more of an issue with best approach and constructing the data properly.
What I am looking to do ...
16
votes
4answers
621 views
How often do you have to roll a 6-sided dice to obtain every number at least once?
I've just played a game with my kids that basically boils down to: whoever rolls every number at least once on a 6-sided dice wins.
I won, eventually, and the others finished 1-2 turns later. Now I'm ...
16
votes
2answers
656 views
Meaning of p-values in regression
When I perform a linear regression in some software packages (for example Mathematica), I get p-values associated with the individual parameters in the model. For, instance the results of a linear ...
15
votes
3answers
289 views
K successes in Bernouli trials, or George Lucas movie experiment
I'm reading "The Drunkard's Walk" now and cannot understand one story from it.
Here it goes:
Imagine that George Lucas makes a new Star Wars film and in one test market decides to perform a crazy ...
15
votes
3answers
304 views
How can I estimate the probability of a random member from one population being “better” than a random member from a different population?
Suppose I have samplings from two distinct populations. If I measure how long it takes each member to do a task, I can easily estimate the mean and variance of each population.
If I now hypothesise ...
15
votes
3answers
382 views
Confidence Interval for variance given one observation
This is a problem from the "7th Kolmogorov Student Olympiad in Probability Theory":
Given one observation $X$ from a $\operatorname{Normal}(\mu,\sigma^2)$ distribution with both parameters ...
15
votes
3answers
3k views
Extending the birthday paradox to more than 2 people
In the traditional birthday paradox the question is "what are the chances that two or more people in a group of n people share a birthday". I'm stuck on a problem which is an extension of this.
...
14
votes
4answers
267 views
The operation of chance in a deterministic world
In Steven Pinker's book Better Angels of Our Nature, he notes that
Probability is a matter of perspective. Viewed at sufficiently close
range, individual events have determinate causes. Even a ...
14
votes
2answers
330 views
Combining classifiers by flipping a coin
I am studying a machine learning course and the lecture slides contain information what I find contradicting with the recommended book.
The problem is the following: there are three classifiers:
...
14
votes
2answers
397 views
Statistics, war stories, data intuition
I think it is fair to say statistics is an applied science so when averages and standard deviations are calculated it is because someone is looking to make some decisions based on those numbers. Part ...
13
votes
2answers
613 views
Can two random variables have the same distribution, yet be almost surely different?
Is it possible that two random variables have the same distribution and yet they are almost surely different?
13
votes
5answers
213 views
Does this quantity related to independence have a name?
Obviously events A and B are independent iff Pr$(A\cap B)$ = Pr$(A)$Pr$(B)$. Let's define a related quantity Q:
$Q\equiv\frac{\mathrm{Pr}(A\cap B)}{\mathrm{Pr}(A)\mathrm{Pr}(B)}$
So A and B are ...
12
votes
5answers
423 views
Is there more to probability than Bayesianism?
As a student in physics, I have experienced the "Why I am a Bayesian" lecture perhaps half a dozen times. It is always the same -- the presenter smugly explains how the Bayesian interpretation is ...
12
votes
4answers
430 views
Do negative probabilities/probability amplitudes have applications outside quantum mechanics?
Quantum Mechanics has generalized probability theory to negative/imaginary numbers, mostly to explain interference patterns, wave/particle duality and generally weird things like that. It can be seen ...
12
votes
3answers
651 views
What is a tight lower bound on the coupon collector time?
In the classic Coupon Collector's problem, it is well known that the time $T$ necessary to complete a set of $n$ randomly-picked coupons satisfies $E[T] \sim n \ln n $,$Var(T) \sim n^2$, and $\Pr(T ...
11
votes
4answers
532 views
Why is statistics useful when many things that matter are one shot things?
I don't know if it's just me, but I am very skeptical of statistics in general. I can understand it in dice games, poker games, etc. Very small, simple, mostly self-contained repeated games are ...
11
votes
1answer
168 views
Deriving Negentropy. Getting stuck
So, this question is somewhat involved but I have painstakingly tried to make it as straight-forward as possible.
Goal: Long story short, there is a derivation of negentropy that does not involve ...
11
votes
1answer
445 views
Probability inequalities
I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts.
My problem is to find an exponential upper ...
10
votes
4answers
1k views
What is so cool about de Finetti's representation theorem?
From Theory of Statistics by Mark J. Schervish (page 12):
Although DeFinetti's representation theorem 1.49 is central to motivating parametric models, it is not actually used in their ...
10
votes
5answers
463 views
Confidence interval and probability - where is the error in this statement?
If someone makes a statement like below:
"Overall, nonsmokers exposed to environmental smoke had a relative
risk of coronary heart disease of 1.25 (95 percent confidence
interval, 1.17 to ...
10
votes
4answers
890 views
What is a purpose of characteristic functions?
I'm hoping that someone can explain, in layman's terms, what a characteristic function is and how it is used in practice. I've read that it is the Fourier transform of the pdf, so I guess I know what ...
10
votes
1answer
779 views
What is the real answer to the Birthday question?
"How large must a class be to make the probability of finding two people with the same birthday at least 50%?"
I have 360 friends on facebook, and, as expected, the distribution of their birthdays is ...
10
votes
2answers
241 views
What's the name of this discrete distribution (recursive difference equation) I derived?
I came across this distribution in a computer game and wanted to learn more about its behaviour. It comes from the decision as to whether a certain event should occur after a given number of player ...
10
votes
3answers
231 views
How do you tell whether good performances come in streaks?
I solve Rubik's cubes as a hobby. I record the time it took me to solve the cube using some software, and so now I have data from thousands of solves. The data is basically a long list of numbers ...
10
votes
1answer
219 views
Is it worthwhile to publish at the refereed wiki StatProb.com?
Background
I read about StatProb.com from a comment on Andrew Gelman's Blog.
According to the website, StatProb is:
StatProb: The Encyclopedia Sponsored
by Statistics and Probability
...
10
votes
1answer
337 views
How to predict when the next event occurs, based on times of previous events?
I'm a high school student and I'm working on a computer programming project, but I don't have a lot of experience in statistics and modeling data beyond a high school statistics course so I'm kinda ...
10
votes
1answer
240 views
How should I mentally deal with Borel's paradox?
I feel a bit uneasy with how I've mentally dealt with Borel's paradox and other associated "paradoxes" dealing with conditional probability. For those who are reading this that aren't familiar with ...
10
votes
3answers
165 views
Build a path probability tree for journeys through a website
I'm currently doing analysis on a website which requires that I create a decision tree diagram showing the likely route that people take whenever they arrive on the website. I am dealing with a ...
10
votes
1answer
488 views
What is this “maximum correlation coefficient”?
A typical image processing statistic is the use of Haralick texture features, which are 14.
I am wondering about the 14th of these features: Given an adjacency map $P$ (which we can simply view an ...
9
votes
3answers
561 views
The prestige magician paradox
You probably know the trick in the movie the prestige :
A magician has found an impressive magic trick : he goes into a machine, close the door, and then disappears and reappears in the other side of ...
9
votes
5answers
929 views
Time taken to hit a pattern of heads and tails in a series of coin-tosses
Inspired by Peter Donnelly's talk at TED, in which he discusses how long it would take for a certain pattern to appear in a series of coin tosses, I created the following script in R. Given two ...
9
votes
3answers
921 views
How to compute the probability associated with absurdly large Z-scores?
Software packages for network motif detection can return enormously high Z-scores (the highest I've seen is 600,000+, but Z-scores of more than 100 are quite common). I plan to show that these ...
