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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170
votes
11answers
45k 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? ...
26
votes
3answers
12k 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 ...
32
votes
8answers
3k 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 ...
32
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 $m$, ...
29
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 ...
28
votes
14answers
28k 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, ...
7
votes
5answers
4k views

Proving Pascal's Rule : ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$

I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$. I suppose I could use the counting rules in probability, perhaps combination= ...
11
votes
4answers
4k 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 ...
9
votes
2answers
2k views

How calculate the probability density function of $Z = X_1/X_2$

Let $X_1$ and $X_2$ be two continuous r.v., my question is: what is the p.d.f of $Z=X_1/X_2$?
13
votes
3answers
4k views

Expectation of the maximum of IID geometric random variables

Given $n$ independent geometric random variables $X_n$, each with probability parameter $p$ (and thus expectation $E\left(X_n\right) = \frac{1}{p}$), what is $$E_n = E\left(\max_{i \in 1 .. ...
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 ...
0
votes
3answers
299 views

Normal Random Variable

I am wondering if I did this question right: Assume that $X$ is a normal random variable. Assume that the expectation is $E[X] = 100$ and the standard deviation is 3. Determine $$ P(E[X] - 6 ...
171
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 ...
4
votes
3answers
2k views

Probability of picking a random natural number

I randomly pick a natural number n. Assuming that I would have picked each number with the same probability, what was the probability for me to pick n before I did it?
14
votes
9answers
2k views

Boy Born on a Tuesday - is it just a language trick?

The following probability question appeared in an earlier thread: I have two children. One is a boy born on a Tuesday. What is the probability I have two boys? The claim was that it is not ...
14
votes
5answers
1k views

Probability of dice sum just greater than 100

Can someone please guide me to a way by which I can solve the following problem. There is a die and 2 players. Rolling stops as soon as some exceeds 100(not including 100 itself). Hence you have the ...
11
votes
1answer
1k views

Probability that a random binary matrix is invertible?

What is the probability that a random $\{0,1\}$, $n \times n$ matrix is invertible? Assume the 0 and 1 are each present in an entry with probability $\frac{1}{2}$. Is there an explicit formula as a ...
17
votes
3answers
779 views

Birthday-coverage problem

I heard an interesting question recently: What is the minimum number of people required to make it more likely than not that all 365 possible birthdays are covered? Monte Carlo simulation ...
15
votes
2answers
689 views

What's the General Expression For Probability of a Failed Gift Exchange Draw

My family does a gift exchange every year at Christmas. There are five couples and we draw names from a hat. If a person draws their own name, or the name of their spouse, all the names go back in a ...
13
votes
2answers
4k views

Probability that a stick randomly broken in two places can form a triangle

Randomly break a stick (or a piece of dry spaghetti, etc.) in two places, forming three pieces. The probability that these three pieces can form a triangle is 1/4 (coordinatize the stick form 0 to 1, ...
11
votes
2answers
9k views

pdf of the difference of two exponentially distributed random variables

Suppose we have $v$ and $u$, both are independent and exponentially distributed random variables with parameters $\mu$ and $\lambda$, respectively. How can we calculate the pdf of $v-u$?
3
votes
2answers
375 views

What is the probability that $x_1+x_2+…+x_n \le n$?

Given that $X_1, X_2...$ are mutually independent random variables. For each $i$ with $1\le i \le n$ the variable $X_i$ is equal to either $0$ or $n+1$ $E(X_i)$ = $1$ also.. if $X_i$ is equal to ...
23
votes
7answers
2k views

Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$

I would like your help with proving that for every $0 \leq k \leq n$, $$\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx . $$ I tried to integration by parts and to get a pattern or to ...
22
votes
2answers
7k 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? ...
5
votes
1answer
983 views

Probability that two random numbers are coprime

This is a really natural question for which I know a stunning solution. So I admit I have a solution, however I would like to see if anybody will come up with something different. The question is ...
18
votes
2answers
1k views

Expected number of rolling a pair of dice to generate all possible sums

A pair of dice is rolled repeatedly until each outcome (2 through 12) has occurred at least once. What is the expected number of rolls necessary for this to occur? Notes: This is not very deep ...
6
votes
4answers
549 views

Why does this expected value simplify as shown?

I was reading about the german tank problem and they say that in a sample of size $k$, from a population of integers from $1,\ldots,N$ the probability that the sample maximum equals $m$ is: ...
3
votes
6answers
196 views

Probability problem

I have $3$ coins, $1$ coin has $2$ heads (HH), 1 coin has $2$ tails (TT), $1$ coin has $1$ head and $1$ tail (HT). I toss the coin, it fells on my hand, and the side i see is a tail. What's the chance ...
2
votes
2answers
373 views

probability distribution of coverage of a set after `X` independently, randomly selected members of the set

I have a set of numbers where I am randomly and independently selecting elements within a set . After a number of these random element selections I want to know the coverage of the elements in the ...
16
votes
6answers
18k views

What is the probability of a coin landing tails 7 times in a row in a series of 150 coin flips?

If you were to flip a coin 150 times, what is the probability that it would land tails 7 times in a row? How about 6 times in a row? Is there some forumula that can calculate this probability?
31
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. ...
13
votes
3answers
1k views

choose a random number between 0 and 1 and record its value. and keep doing it until the sum of the numbers exceeds 1. how many tries?

choose a random number between 0 and 1 and record its value. Do this again and add the second number to the first number. Keep doing this until the sum of the numbers exceeds 1. What's the expected ...
11
votes
3answers
594 views

A question about Poker (and probability in general)

Okay, so I've been thinking about this question for a long time, and I'm starting to think that there isn't an answer. So please read the question, and if there is an answer, tell how you came to it, ...
4
votes
2answers
2k views

How can I (algorithmically) count the number of ways n m-sided dice can add up to a given number?

I am trying to identify the general case algorithm for counting the different ways dice can add to a given number. For instance, there are six ways to roll a seven with two 6-dice. I've spent quite ...
1
vote
4answers
319 views

Probability Problem on Divisibility of Sum by 3

From the 3-element subsets of $\{1, 2, 3, \ldots , 100\}$ (the set of the first 100 positive integers), a subset $(x, y, z)$ is picked randomly. What is the probability that $x + y + z$ is divisible ...
7
votes
2answers
2k views

If n balls are thrown into k bins, what is the probability that every bin gets at least one ball?

If $n$ balls are thrown into $k$ bins (uniformly at random and independently), what is the probability that every bin gets at least one ball? i.e. If we write $X$ for the number of empty bins, what ...
4
votes
1answer
652 views

how to show convergence in probability imply convergence a.s. in this case?

Assume that $X_1,\cdots,X_n$ are independent r.v., not necessarily iid, Let $S_n=X_1+\cdots+X_n$, suppose that $S_n$ converges in probability, how can we show that $S_n$ converges a.s.?
10
votes
2answers
2k views

Expected number of tosses for two coins to achieve the same outcome for five consecutive flips

Consider two unbiased coins. Toss both until last 5 sequence outcome are same. That means we stop when output of the sequence of both are as follows: HTTHTHHTH , HHTTTHHTH. What is the expected ...
23
votes
7answers
4k views

If a 1 meter rope is cut at two uniformly randomly chosen points (to give three pieces), what is the average length of the smallest piece?

I got this question as a mathematical puzzle from a friend. It looks similar to this one ...
20
votes
3answers
1k views

coin toss question

Two players A and B each has a fair coin and they start to toss simultaneously (counted as one round). They toss in $n$ ($\ge 1$) rounds and stop because they have accumulated the same number of heads ...
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 ...
17
votes
4answers
1k views

probability textbooks

Has anyone compiled a moderately comprehensive list on the web or elsewhere of textbooks on probability For students who have not been introduced to the subject before That introduce both discrete ...
6
votes
1answer
7k views

Calculation of the n-th central moment of the normal distribution $\mathcal{N}(\mu,\sigma^2)$

Since integration is not my strong suit I need some feedback on this, please: Let $Y$ be $\mathcal{N}(\mu,\sigma^2)$, the normal distrubution with parameters $\mu$ and $\sigma^2$. I know $\mu$ is the ...
4
votes
1answer
182 views

Variance of time to find first duplicate

In repeated uniform sampling from $\{1,\dots,n\}$ the mean time $E(X)$ to find the first duplicate is asymptotically $\sqrt{n\pi/2}$. What about the variance? The variance is $E(X^2) -E(X)^2$. ...
0
votes
3answers
2k views

What are the odd of a single coin toss after many consecutive ones?

I have little knowledge of probability and have recently been thinking of a math problem I am sure you guys could answer If for example I tossed a coin ten times and got all heads, would the odds of ...
5
votes
4answers
8k views

Expected value of maximum of two random variables from uniform distribution

If I have two variables $X$ and $Y$ which randomly take on values uniformly from the range $[a,b]$ (all values equally probable), what is the expected value for $\max(X,Y)$?
4
votes
1answer
802 views

Expected number of draws until the first good element is chosen

A population has $G$ good and $B$ bad elements, $G+B=N$. Elements are drawn one by one at random without replacement. Suppose the first good element appears on draw number $X$. Find a simple formula, ...
3
votes
2answers
541 views

Combinatorics Distribution - Number of integer solutions Concept Explanation

I reading my textbook and I don't understand the concept of distributions or number of solutions to an equation. It's explained that this problem is 1/4 types of sampling/distributions problems. An ...
3
votes
2answers
7k views

Sum of independent Gamma distributions is a Gamma distribution

If $X\sim \mathrm{Gamma}(a_1,b)$ and $Y \sim \mathrm{Gamma}(a_2,b)$, I need to prove $X+Y\sim(a_1+a_2,b)$ if $X$ and $Y$ are independent. I am trying to apply formula for independence integral and ...
16
votes
8answers
4k views

Is it possible to 'split' coin flipping 3 ways?

When flipping a coin to make important decisions in life you can flip once to choose between 2 possible outcomes. (Heads I eat cake, Tails I eat chocolate!) You can also flip twice to choose between ...