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Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. 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.

289
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6answers
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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? ...
284
votes
5answers
65k views

In Russian roulette, is it best to go first?

Assume that we are playing a game of Russian roulette (6 chambers). Assume 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 ...
254
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14answers
35k 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 ...
206
votes
17answers
34k 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: ...
179
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25answers
15k 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 ...
165
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30answers
27k views

Counterintuitive examples in probability

I want to teach a short course in probability and I am looking for some counter-intuitive examples for it. Results that seems to be obviously false but they true or vice versa... I already found some ...
147
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15answers
47k views

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, ...
145
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4answers
8k 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 ...
134
votes
6answers
22k 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 ...
118
votes
4answers
17k 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 ...
114
votes
8answers
8k 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 ...
112
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13answers
62k 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 ...
111
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8answers
5k views

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. ...
106
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11answers
163k 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.
101
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9answers
28k 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, ...
95
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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 ...
91
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19answers
21k 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) ...
91
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4answers
9k 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 ...
90
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5answers
12k 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, ...
88
votes
10answers
7k 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
5answers
132k 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?
86
votes
10answers
71k 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?
86
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14answers
19k 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 ...
83
votes
9answers
17k views

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 ...
83
votes
18answers
176k 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, ...
83
votes
3answers
72k 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 ...
82
votes
2answers
45k 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? ...
80
votes
7answers
428k views

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 ...
77
votes
12answers
19k 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?
77
votes
5answers
41k 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 ...
76
votes
21answers
15k 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 ...
75
votes
3answers
16k views

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 ...
74
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4answers
7k 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 ...
72
votes
8answers
12k 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 ...
71
votes
6answers
4k 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 ...
70
votes
4answers
98k 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 ...
70
votes
3answers
10k 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 ...
66
votes
6answers
9k 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"...
65
votes
10answers
14k 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 ...
64
votes
6answers
17k 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 ...
64
votes
6answers
6k views

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 ...
64
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4answers
108k 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?
63
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9answers
15k 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{(...
63
votes
8answers
35k views

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 ...
62
votes
10answers
36k 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 ...
61
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3answers
6k 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 ...
60
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20answers
13k views

Coin flipping probability game ; 7 flips vs 8 flips

Your friend flips a coin 7 times and you flip a coin 8 times; the person who got the most tails wins. If you get an equal amount, your friend wins. There is a 50% chance of you winning the game and a ...
60
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12answers
17k views

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 ...
60
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4answers
3k views

What's the probability that a sequence of coin flips never has twice as many heads as tails?

I gave my friend this problem as a brainteaser; while her attempted solution didn't work, it raised an interesting question. I flip a fair coin repeatedly and record the results. I stop as soon as ...
60
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3answers
2k views

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}\...