This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under ...

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184
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5answers
36k 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 ...
173
votes
14answers
22k views

Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet? [closed]

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 ...
172
votes
11answers
46k views

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? ...
122
votes
24answers
10k 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 no coins were in reach. There was however an SD card on my desk: Given that I don't know the bias of this SD card, would flipping it be ...
91
votes
20answers
18k 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, ...
87
votes
8answers
3k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (March. 2014) This question has been moved to mathoverflow; see here. Randomly break a stick in five places. Question: What is the probability that the resulting six pieces can form a ...
84
votes
7answers
6k 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 ...
83
votes
5answers
7k 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 ...
69
votes
20answers
17k 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) far less likely ...
61
votes
8answers
14k 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?
58
votes
7answers
6k 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 ...
55
votes
6answers
4k 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 ...
53
votes
3answers
5k 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 ...
50
votes
3answers
3k 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 ...
49
votes
19answers
11k 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 ...
46
votes
6answers
26k 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 ...
42
votes
5answers
5k 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 ...
40
votes
5answers
7k views

Convergence of $np(n)$ where $p(n)=\sum_{j=\lceil n/2\rceil}^{n-1} {p(j)\over j}$

Some years ago I was interested in the following Markov chain whose state space is the positive integers. The chain begins at state "1", and from state "n" the chain next jumps to a state uniformly ...
39
votes
4answers
2k 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 ...
39
votes
8answers
13k views

How to determine if coin comes up heads more often than tails?

Not a math student, so forgive me if the question seems trivial or if I pose it "wrong". Here goes... Say I'm flipping a coin a n times. I am not sure if it's a "fair" coin, meaning I am not sure if ...
38
votes
9answers
4k views

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

I'm a very beginner in mathematics and there is one thing I've been wondering recently. The formula for the normal distribution is: ...
38
votes
1answer
982 views

Does a randomly chosen series diverge?

Pick a point at random in the interval $[0,1]$, call it $P_1$. Pick another point at random in the interval $[0,P_1]$, call it $P_2$. Pick another point at random in the interval $[0,P2]$, call it ...
37
votes
1answer
1k views

Identity for simple 1D random walk

The question is to find a purely probabilistic proof of the following identity, valid for every integer $n\geqslant1$, where $(S_n)_{n\geqslant0}$ denotes a standard simple random walk: $$ ...
37
votes
5answers
1k views

Solutions to $\binom{n}{5} = 2 \binom{m}{5}$

In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says: On National Public Radio, the Weekend Edition program posed the following probability problem: Given a certain number of ...
37
votes
1answer
895 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 ...
35
votes
9answers
4k 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 ...
35
votes
6answers
2k views

A variant of the Monty Hall problem

Everybody knows the famous Monty Hall problem; way too much ink has been spilled over it already. Let's take it as a given and consider the following variant of the problem that I thought up this ...
34
votes
5answers
4k views

Free throw interview question

I recently had an interview question that posed the following... Suppose you are shooting free throws and each shot has a 60% chance of going in (there is no "learning" effect and "depreciation" ...
34
votes
4answers
4k views

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 $m$, ...
33
votes
2answers
1k views

Is there a clever solution to this elementary probability/combinatorics problem?

My friend was asked the following problem in an interview a while back, and it has a nice answer, leading me to believe that there is an equally nice solution. Suppose that there are 42 bags, ...
33
votes
4answers
797 views

Strange Patience Game

I read about this game as a kid, but my maths was never up to solving it: The score starts at zero. Take a shuffled pack of cards and keep dealing face up until you reach the first Ace, at which the ...
32
votes
3answers
14k 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. ...
32
votes
4answers
1k views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
32
votes
3answers
1k views

Inequality for expected value

A colleague popped into my office this afternoon and asked me the following question. He told me there is a clever proof when $n=2$. I couldn't do anything with it, so I thought I'd post it here and ...
31
votes
3answers
6k views

Making 400k random choices from 400k samples seems to always end up with 63% distinct choices, why?

I have a very simple simulation program, the sequence is: Create an array of 400k elements Use a PRNG to pick an index, and mark the element (repeat 400k times) Count number of marked elements. An ...
30
votes
7answers
4k 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 ...
30
votes
3answers
4k views

Why does this not seem to be random?

I was running a procedure to be like one of those games were people try to guess a number between 1 and 100 where there are 100 people guessing.I then averaged how many different guesses there are. ...
30
votes
1answer
2k views

Minesweeper - Chance of one-click win

I'd like to know if it's possible to calculate the odds of winning a game of Minesweeper (on easy difficulty) in a single click. This page documents a bug that occurs if you do so, and they calculate ...
29
votes
5answers
6k views

Chance of getting a date

A girl you like tells you that there's a 1% chance she'll go out on a date with you. What is the expected number of times you have to ask her out before you get a date? (Not a homework problem - I ...
29
votes
2answers
4k views

Why is the error function defined as it is?

$\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$ Of ...
29
votes
1answer
894 views

What is the expected number of steps in the following process?

We have $n$ boxes. And initially there are $x_1, x_2, x_3, \ldots, x_n$ marbles in each box. We randomly (with equal probabilities) select one of the boxes. We take one marble from it and we put it ...
28
votes
14answers
29k 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, ...
28
votes
5answers
2k views

How to find a random axis or unit vector in 3D?

I would like to generate a random axis or unit vector in 3D. In 2D it would be easy, I could just pick an angle between 0 and 2*Pi and use the unit vector pointing in that direction. But in 3D I ...
28
votes
10answers
8k views

Odds of winning at minesweeper with perfect play

How would someone go about doing this? Assume that the first "click" will never be a bomb, and that the number of mines and the area are both known. Rather hoping there is a clever way to do this, but ...
28
votes
3answers
2k views

Why is this coin-flipping probability problem unsolved?

You play a game flipping a fair coin. You may stop after any trial, at which point you are paid in dollars the percentage of heads flipped. So if on the first trial you flip a head, you should stop ...
27
votes
2answers
13k 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 ...
26
votes
5answers
3k 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, ...
26
votes
7answers
31k 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.
26
votes
5answers
883 views

A probabilistic game with balls and urns

We have $N$ urns forming a circle and $M<N$ balls (for example, $N=9,M=6$ in the diagram). In each step we visit, sequentially, clockwise, an urn. If it's occupied by a ball, we say a $hit$ has ...
25
votes
7answers
2k views

New to probability - Is this true?

I read this somewhere and can't find it to verify: "If you don't have a dice with 12 sides, but do have one with 20 sides and you need to make rolls for 12 sides, you can use the numbers 1-12 on the ...