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.
86,871
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305
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7answers
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Multiple-choice question about the probability of a random answer to itself being correct
I found this math "problem" on the internet, and I'm wondering if it has an answer:
Question: If you choose an answer to this question at random, what is the probability that you will be correct?
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297
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5answers
70k views
In Russian roulette, is it best to go first?
Assume that we are playing a game of Russian roulette (6 chambers) and that there is no shuffling after the shot is fired.
I was wondering if you have an advantage in going first?
If so, how big of ...
263
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13answers
37k views
Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet?
Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet."
I have a (not very mathematically intelligent) friend who ...
216
votes
18answers
36k views
Do men or women have more brothers?
Do men or women have more brothers?
I think women have more as no man can be his own brother. But how one can prove it rigorously?
I am going to suggest some reasonable background assumptions:
...
185
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25answers
16k views
Can a coin with an unknown bias be treated as fair?
This morning, I wanted to flip a coin to make a decision but only had an SD card:
Given that I don't know the bias of this SD card, would flipping it be considered a "fair toss"?
I thought if I'm ...
182
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31answers
32k views
Counterintuitive examples in probability
I want to teach a short course in probability and I am looking for some counter-intuitive examples in Probability. The results seem to be obviously false but they true or vice versa.
I already found ...
155
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15answers
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Monty hall problem extended.
I just learned about the Monty Hall problem and found it quite amazing. So I thought about extending the problem a bit to understand more about it.
In this modification of the Monty Hall Problem, ...
151
votes
4answers
9k views
Sum of random decreasing numbers between 0 and 1: does it converge??
Let's define a sequence of numbers between 0 and 1. The first term, $r_1$ will be chosen uniformly randomly from $(0, 1)$, but now we iterate this process choosing $r_2$ from $(0, r_1)$, and so on, so ...
149
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17answers
84k views
Taking Seats on a Plane
This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless
Imagine there are a 100 people in line to ...
142
votes
6answers
28k views
What's 4 times more likely than 80%?
There's an 80% probability of a certain outcome, we get some new information that means that outcome is 4 times more likely to occur.
What's the new probability as a percentage and how do you work it ...
138
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11answers
204k views
What is the best book to learn probability?
Question is quite straight... I'm not very good in this subject but need to understand at a good level.
130
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5answers
177k views
Is the product of two Gaussian random variables also a Gaussian?
Say I have $X \sim \mathcal N(a, b)$ and $Y\sim \mathcal N(c, d)$. Is $XY$ also normally distributed?
Is the answer any different if we know that $X$ and $Y$ are independent?
127
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4answers
20k views
What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?
My friend gave me this puzzle:
What is the probability that a point chosen at random from the interior of an equilateral triangle is closer to the center than any of its edges?
I tried to draw ...
119
votes
8answers
9k views
probability $2/4$ vs $3/6$
Recently I was asked the following in an interview:
If you are a pretty good basketball player, and were betting on whether you could make $2$ out of $4$ or $3$ out of $6$ baskets, which would you ...
114
votes
6answers
16k views
Intuition behind Conditional Expectation
I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, ...
114
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9answers
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Probability that a stick randomly broken in five places can form a tetrahedron
Edit (June. 2015) This question has been moved to MathOverflow, where a recent write-up finds a similar approximation as leonbloy's post below; see here.
Randomly break a stick in five places.
...
110
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9answers
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What is the difference between independent and mutually exclusive events?
Two events are mutually exclusive if they can't both happen.
Independent events are events where knowledge of the probability of one doesn't change the probability of the other.
Are these ...
109
votes
4answers
158k views
What is the difference and relationship between the binomial and Bernoulli distributions?
How should I understand the difference or relationship between binomial and Bernoulli distribution?
109
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3answers
99k views
Expected time to roll all 1 through 6 on a die
What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die? The order in which the numbers appear does not matter.
I had this questions explained ...
107
votes
5answers
60k views
Could someone explain conditional independence?
My understanding right now is that an example of conditional independence would be:
If two people live in the same city, the probability that person A gets home in time for dinner, and the ...
106
votes
9answers
32k views
If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?
Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair?
Furthermore, ...
103
votes
20answers
231k views
In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
I just dipped into a book, The Drunkard's Walk - How Randomness Rules Our Lives, ...
99
votes
8answers
10k views
Is Bayes' Theorem really that interesting?
I have trouble understanding the massive importance that is afforded to Bayes' theorem in undergraduate courses in probability and popular science.
From the purely mathematical point of view, I think ...
98
votes
11answers
11k views
Mathematician vs. Computer: A Game
A mathematician and a computer are playing a game: First, the mathematician chooses an integer from the range $2,...,1000$. Then, the computer chooses an integer uniformly at random from the same ...
95
votes
19answers
22k views
Should I put number combinations like 1111111 onto my lottery ticket?
Suppose the winning combination consists of $7$ digits, each digit randomly ranging from $0$ to $9$. So the probability of $1111111$, $3141592$ and $8174249$ are the same. But $1111111$ seems (to me) ...
95
votes
4answers
11k views
The “pepperoni pizza problem”
This problem arose in a different context at work, but I have translated it to pizza.
Suppose you have a circular pizza of radius $R$. Upon this disc, $n$ pepperoni will be distributed completely ...
94
votes
10answers
92k views
Given a die, what is the probability that the second roll of a die will be less than the first roll?
If you are given a die and asked to roll it twice. What is the probability that the value of the second roll will be less than the value of the first roll?
94
votes
2answers
58k views
Expectation of the maximum of gaussian random variables
Is there an exact or good approximate expression for the expectation, variance or other moments of the maximum of $n$ independent, identically distributed gaussian random variables where $n$ is large?
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92
votes
12answers
24k views
Why do bell curves appear everywhere?
Why do most probability graphs show a bell curve? I've been wondering why... Is it just something natural, like the fibonacci sequence?
91
votes
14answers
23k views
I roll a die repeatedly until I get 6, and then count the number of 3s I got. What's my expected number of 3s?
Consider the following experiment. I roll a die repeatedly until the die returns 6, then I count the number of times 3 appeared in the random variable $X$. What is $E[X]$?
Thoughts: I expect to roll ...
88
votes
10answers
8k views
Would you ever stop rolling the die? [duplicate]
You have a six-sided die. You keep a cumulative total of your dice rolls. (E.g. if you roll a 3, then a 5, then a 2, your cumulative total is 10.) If your cumulative total is ever equal to a perfect ...
87
votes
3answers
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Why did no student correctly find a pair of $2\times 2$ matrices with the same determinant and trace that are not similar?
I gave the following problem to students:
Two $n\times n$ matrices $A$ and $B$ are similar if there exists a nonsingular matrix $P$ such that $A=P^{-1}BP$.
Prove that if $A$ and $B$ are two ...
85
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10answers
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Expected number of unpecked chicks - NYT article
In this article, the winner of the math competition answered this question correctly:
In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to ...
82
votes
4answers
109k views
Probability density function vs. probability mass function
I've a confession to make. I've been using pdf's and pmf's without actually knowing what they are. The idea that I've been having so long is that density = area under the curve but if I look at it ...
81
votes
21answers
22k views
How to generate a random number between 1 and 10 with a six-sided die?
Just for fun, I am trying to find a good method to generate a random number between 1 and 10 (uniformly) with an unbiased six-sided die.
I found a way, but it may requires a lot of steps before ...
79
votes
7answers
5k views
Rigorous nature of combinatorics
Context: I'm a high school student, who has only ever had an introductory treatment, if that, on combinatorics. As such, the extent to which I have seen combinatoric applications is limited to ...
77
votes
14answers
116k views
Expected Number of Coin Tosses to Get Five Consecutive Heads
A fair coin is tossed repeatedly until 5 consecutive heads occurs.
What is the expected number of coin tosses?
77
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4answers
9k views
Why did my friend lose all his money?
Not sure if this is a question for math.se or stats.se, but here we go:
Our MUD (Multi-User-Dungeon, a sort of textbased world of warcraft) has a casino where players can play a simple roulette.
My ...
76
votes
8answers
14k views
What is the chance to get a parking ticket in half an hour if the chance to get a ticket is 80% in 1 hour?
This sounds more like a brain teaser, but I had some kink to think it through :( Suppose you're parking at a non-parking zone, the probability to get a parking ticket is 80% in 1 hour, what is the ...
76
votes
12answers
21k views
The Monty Hall problem
I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the ...
72
votes
4answers
43k views
Intuition behind using complementary CDF to compute expectation for nonnegative random variables
I've read the proof for why $\int_0^\infty P(X >x)dx=E[X]$ for nonnegative random variables (located here) and understand its mechanics, but I'm having trouble understanding the intuition behind ...
72
votes
3answers
11k views
Mathematical research of Pokémon
In competitive Pokémon-play, two players pick a team of six Pokémon out of the 718 available. These are picked independently, that is, player $A$ is unaware of player $B$'s choice of Pokémon. Some ...
70
votes
4answers
8k views
Drunk man with a set of keys.
I found this problem in a contest of years ago, but I'm not very good at probability, so I prefer to see how you do it:
A man gets drunk half of the days of a month. To open his house, he has a set ...
70
votes
2answers
2k views
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 ...
69
votes
6answers
19k views
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 ...
69
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9answers
18k views
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{(...
69
votes
6answers
46k views
Understanding Borel sets
I'm studying Probability theory, but I can't fully understand what are Borel sets. In my understanding, an example would be if we have a line segment [0, 1], then a Borel set on this interval is a set ...
68
votes
6answers
10k views
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"...
67
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9answers
87k views
Probability of 3 people in a room of 30 having the same birthday
I have been looking at the birthday problem (http://en.wikipedia.org/wiki/Birthday_problem) and I am trying to figure out what the probability of 3 people sharing a birthday in a room of 30 people is. ...
65
votes
10answers
38k views
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 ...
