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 ...
131
votes
10answers
28k 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?
...
19
votes
3answers
5k 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 ...
21
votes
4answers
1k 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 ...
23
votes
4answers
1k 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 $m$, ...
23
votes
10answers
14k 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, ...
6
votes
2answers
920 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 $Z=X_1/X_2$?
7
votes
4answers
1k 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 ...
20
votes
6answers
1k 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 ...
12
votes
9answers
1k 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 ...
48
votes
6answers
3k 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 ...
16
votes
3answers
572 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 ...
5
votes
5answers
2k views
Proving ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ 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= ...
3
votes
3answers
1k 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?
11
votes
3answers
544 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, ...
9
votes
3answers
2k 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 .. ...
14
votes
2answers
526 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 ...
8
votes
1answer
742 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 ...
5
votes
1answer
756 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 ...
2
votes
1answer
199 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 ...
138
votes
14answers
17k 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 ...
11
votes
5answers
8k 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?
19
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 ...
12
votes
4answers
773 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 ...
10
votes
2answers
3k 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?
...
16
votes
2answers
801 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 ...
10
votes
2answers
2k 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, ...
3
votes
2answers
729 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 ...
5
votes
4answers
350 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:
...
7
votes
3answers
264 views
Expectation of an event
Let $A$ be an array of length 1000 with all entries 0. I want to fill up $A$ with ones using the following approach:
At each iteration I take three random integers $(j_1,j_2,j_3)$ from [1,1000] with ...
4
votes
1answer
295 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.?
1
vote
4answers
118 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 ...
23
votes
2answers
6k 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. ...
10
votes
2answers
982 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 ...
14
votes
3answers
611 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 ...
15
votes
3answers
1k views
Consecutive birthdays probability
Let $n$ be a number of people. At least two of them may be born on the same day of the year with probability: $$1-\prod_{i=0}^{n-1} \frac{365-i}{365}$$
But what is the probability that at least two ...
5
votes
2answers
584 views
Time until a consecutive sequence of ones in a random bit sequence
This a reformulation of a practical problem I encountered.
Say we have an infinite sequence of random, i.i.d bits. For each bit $X_i$, $P(X_i=1)=p$.
What is the expected time until we get a sequence ...
8
votes
2answers
899 views
Algebra of Random Variables?
I've been looking online (and in teaching journals) for a good introduction to Algebras of Random Variables (on an undergraduate level) and their usage, and have come up short. I know I can find the ...
4
votes
3answers
659 views
a question related to two competing patterns in coin tossing
If I have a two-sided coin with probability $p$ showing head. I repeatedly toss it until either HTHTH or HTHH appears. Can you calculate
1) the probability when I got HTHTH, and
2) the expected ...
8
votes
5answers
559 views
Proof that $x \Phi(x) + \Phi'(x) \geq 0$ $\forall x$, where $\Phi$ is the normal CDF
As title. Can anyone supply a simple proof that
$$x \Phi(x) + \Phi'(x) \geq 0 \quad \forall x\in\mathbb{R}$$
where $\Phi$ is the standard normal CDF, i.e.
$$\Phi(x) = \int_{-\infty}^x ...
3
votes
4answers
1k views
Distribution of Functions of Random Variables
In general, how would one find the distribution of $f(X)$ where $X$ is a random variable? Or consider the inverse problem of finding the distribution of $X$ given the distribution of $f(X)$. For ...
0
votes
1answer
77 views
Probability of choosing C out of T students with restrictions
There is a class of $T$ students, consisting of $G$ girls and $(T - G)$ boys. Out of the $T$ students, only $C$ are selected for an examination. What is the probability that there are at least $K$ ...
-2
votes
1answer
238 views
Functions of Random Variables
Let X be a discrete random variable with probability function fX .
Find formulas for the probability function and the distribution function of
Y =(X − a)^2, where a is an arbitrary constant.
4
votes
1answer
451 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, ...
2
votes
1answer
205 views
Card probability problem [duplicate]
Possible Duplicate:
Probability problem
I found the following problem in Rosen's Discrete Mathematics and Its Applications 6th ed.:
There are three cards in a box. Both sides of one card ...
2
votes
2answers
895 views
Expectation of number of trials before success in an urn problem without replacement [duplicate]
Possible Duplicate:
Expected number of draws until the first good element is chosen
An urn contains $b$ blue balls and $r$ red balls. Balls are removed at random without replacement until ...
2
votes
1answer
301 views
Alternative Expected Value Proof
I am currently tasked with proving an alternative definition of the expected value function.
Considering X to be a random variable that takes all positive integers, I have to prove that
...
1
vote
2answers
84 views
Probability of getting >70% in exam with 50 yes/no questions
In a paper containing 50 yes/no questions, I am trying to find the probability of getting 70%.
Using binomial distribution,
$$P(X\ge70\%)=\sum_{k=25}^{50} \binom{50}{k}\left(\frac{1}{2}\right)^{50}$$
...
1
vote
2answers
196 views
Variance of discrete random variables
Two fair and independent dice (each with six faces) are thrown. Let $X_1$ be the score on the first die and $X_2$ the score on the second. Let $X = X_1 + X_2$ , $Y = X_1 X_2$ and $Z = \min(X_1; ...
1
vote
1answer
151 views
Can you determine a formula for this problem?
Given: A list of integers is there.Now there are 2 buckets -bucket A and bucket B.This step is repeated as long as there are numbers left in the list.Integers from start or end of the list are ...
1
vote
2answers
397 views
Probability Density Function Validity
If X is a continuous random variable with range $[x_l,\infty)$ and p.d.f.
$f_x(X) \propto x^{-a}$, for $x\in[x_l,\infty)$
for some values $x_l > 0$ and $a \in \mathbb{R}$.
How do I calculate the ...
