Questions tagged [probability]

A probability provides a quantitative description of the likely occurrence of a particular event.

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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 ...
67k views

Why does a 95% Confidence Interval (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 ...
77k views

Can a probability distribution value exceeding 1 be OK?

On the Wikipedia page about naive Bayes classifiers, there is this line: $p(\mathrm{height}|\mathrm{male}) = 1.5789$ (A probability distribution over 1 is OK. It is the area under the bell curve ...
69k views

Amazon interview question—probability of 2nd interview

I got this question during an interview with Amazon: 50% of all people who receive a first interview receive a second interview 95% of your friends that got a second interview felt they had a good ...
23k views

Why do we need sigma-algebras to define probability spaces?

We have a random experiment with different outcomes forming the sample space $\Omega,$ on which we look with interest at certain patterns, called events $\mathscr{F}.$ Sigma-algebras (or sigma-fields) ...
24k views

At each step of a limiting infinite process, put 10 balls in an urn and remove one at random. How many balls are left?

The question (slightly modified) goes as follows and if you have never encountered it before you can check it in example 6a, chapter 2, of Sheldon Ross' A First Course in Probability: Suppose that ...
83k views

Numerical example to understand Expectation-Maximization

I am trying to get a good grasp on the EM algorithm, to be able to implement and use it. I spent a full day reading the theory and a paper where EM is used to track an aircraft using the position ...
78k views

What's the difference between probability and statistics?

What's the difference between probability and statistics, and why are they studied together?
50k views

If 900 out of 1000 people say a car is blue, what is the probability that it is blue?

This initially arose in connection some work we are doing to a model to classify natural text, but I've simplified it... Perhaps too much. You have a blue car (by some objective scientific measure - ...
15k views

If I have a 58% chance of winning a point, what's the chance of me winning a ping pong game to 21, win by 2?

I have a bet with a co-worker that out of 50 ping pong games (first to win 21 points, win by 2), I will win all 50. So far we've played 15 games and on average I win 58% of the points, plus I've won ...
68k views

What does a “closed-form solution” mean?

I have come across the term "closed-form solution" quite often. What does a closed-form solution mean? How does one determine if a close-form solution exists for a given problem? Searching online, I ...
15k views

Probability of a single real-life future event: What does it mean when they say that “Hillary has a 75% chance of winning”?

As the election is a one time event, it is not an experiment that can be repeated. So exactly what does the statement "Hillary has a 75% chance of winning" technically mean? I am seeking a ...
8k views

Is there any *mathematical* basis for the Bayesian vs frequentist debate?

It says on Wikipedia that: the mathematics [of probability] is largely independent of any interpretation of probability. Question: Then if we want to be mathematically correct, shouldn't we ...
29k 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 ...
23k views

Why to optimize max log probability instead of probability

In most machine learning tasks where you can formulate some probability $p$ which should be maximised, we would actually optimize the log probability $\log p$ instead of the probability for some ...
9k views

Is this chart showing the likelihood of a terrorist attack statistically useful?

I'm seeing this image passed around a lot. I have a gut-feeling that the information provided this way is somehow incomplete or even erroneous, but I'm not well versed enough in statistics to respond....
4k views

How much to pay? A practical problem

This is not a home work question but real problem faced by our company. Very recently (2 days ago) we ordered for manufacturing of 10000 product labels to a dealer. Dealer is independent person. He ...
9k views

Brain teaser: How to generate 7 integers with equal probability using a biased coin that has a pr(head) = p?

This is a question I found on Glassdoor: How does one generate 7 integers with equal probability using a coin that has a $\mathbb{Pr}(\text{Head}) = p\in(0,1)$? Basically, you have a coin that may or ...
37k views

Does 10 heads in a row increase the chance of the next toss being a tail?

I assume the following is true: assuming a fair coin, getting 10 heads in a row whilst tossing a coin does not increase the chance of the next coin toss being a tail, no matter what amount of ...
13k 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 ...
45k views

How to generate correlated random numbers (given means, variances and degree of correlation)?

I'm sorry if this seems a bit too basic, but I guess I'm just looking to confirm understanding here. I get the sense I'd have to do this in two steps, and I've started trying to grok correlation ...
8k 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 ...
12k views

How to tell the probability of failure if there were no failures?

I was wondering if there is a way to tell the probability of something failing (a product) if we have 100,000 products in the field for 1 year and with no failures? What is the probability that one of ...
41k views

Expected number of ratio of girls vs boys birth

I have came across a question in job interview aptitude test for critical thinking. It is goes something like this: The Zorganian Republic has some very strange customs. Couples only wish to have ...
11k 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 ...
7k views

Does the reciprocal of a probability represent anything?

I was wondering if the reciprocal of P(X = 1) represents anything in particular?
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Are CDFs more fundamental than PDFs?

My stat prof basically said, if given one of the following three, you can find the other two: Cumulative distribution function Moment Generating Function Probability Density Function But my ...
18k 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" ...
17k views

How often do you have to roll a 6-sided die 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 die wins. I won, eventually, and the others finished 1-2 turns later. Now I'm ...
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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, who ...
6k views

Would a Bayesian admit that there is one fixed parameter value?

In Bayesian data analysis, parameters are treated as random variables. This stems from the Bayesian subjective conceptualization of probability. But do Bayesians theoretically acknowledge that there ...
5k 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 ...
50k views

PP-plots vs. QQ-plots

What is the difference between probability plots, PP-plots and QQ-plots when trying to analyse a fitted distribution to data?
26k views

Bayesian vs frequentist Interpretations of Probability

Can someone give a good rundown of the differences between the Bayesian and the frequentist approach to probability? From what I understand: The frequentists view is that the data is a repeatable ...
12k views

Help me calculate how many people will come to my wedding! Can I attribute a percentage to each person and add them?

I am planning my wedding. I wish to estimate how many people will come to my wedding. I have created a list of people and the chance that they will attend in percentage. For example ...
12k views

What is the 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 ...
4k views

Will the fact that my Italian son is going to attend a primary school change the expected number of Italian children to be present in his class?

This is a question stemming from a real-life situation, for which I have been genuinely puzzled about its answer. My son is due to start primary school in London. As we are Italian, I was curious to ...
2k 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 ...
13k views

Why does “explaining away” make intuitive sense?

I recently learned about a principle of probabilistic reasoning called "explaining away," and I am trying to grasp an intuition for it. Let me set up a scenario. Let $A$ be the event that an ...
20k views

Probability distribution for different probabilities

If I wanted to get the probability of 9 successes in 16 trials with each trial having a probability of 0.6 I could use a binomial distribution. What could I use if each of the 16 trials has a ...
14k 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.
8k views

Random walk on the edges of a cube

An ant is placed in a corner of a cube and cannot move. A spider starts from the opposite corner, and can move along the cube's edges in any direction $(x,y,z)$ with equal probability $1/3$. On ...
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Birthday paradox with a (huge) twist: Probability of sharing exact same date of birth with partner?

I share the same birthdate as my boyfriend, same date but also same year, our births are seperated by merely 5 hours or so. I know that the chances of meeting someone who was born on the same date ...
16k views

Generic sum of Gamma random variables

I have read that the sum of Gamma random variables with the same scale parameter is another Gamma random variable. I've also seen the paper by Moschopoulos describing a method for the summation of a ...
8k views

Are there any examples of where the central limit theorem does not hold?

Wikipedia says - In probability theory, the central limit theorem (CLT) establishes that, in most situations, when independent random variables are added, their properly normalized sum tends ...
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Should I teach Bayesian or frequentist statistics first?

I am helping my boys, currently in high school, understanding statistics, and I am considering beginning with some simple examples without disregarding some glimpses to theory. My goal would be to ...
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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 ...
2k views

If I generate a random symmetric matrix, what's the chance it is positive definite?

I got a strange question when I was experimenting some convex optimizations. The question is: Suppose I randomly (say standard normal distribution) generate a $N \times N$ symmetric matrix, (for ...
I am interested in the following one-sided Cantelli's version of the Chebyshev inequality: $$\mathbb P(X - \mathbb E (X) \geq t) \leq \frac{\mathrm{Var}(X)}{\mathrm{Var}(X) + t^2} \,.$$ Basically, ...