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Questions tagged [probability]

For questions about probability. independence, total probability and conditional probability. For questions about the theoretical footing of probability use [tag:probability-theory]. For questions about specific probability distributions, use [tag:probability-distributions].

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134 votes
3 answers

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 ...
eternalmatt's user avatar
  • 1,505
58 votes
3 answers

Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$

Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show, $$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$ when $X$ has : a) a discrete distribution, b) a continuous ...
Jon Gan's user avatar
  • 1,521
31 votes
1 answer

Probability distribution in the coupon collector's problem

I'm trying to solve the well known Coupon Collector's Problem by explicitly finding the probability distribution (so far all the methods I read involve using some sort of trick). However, I'm not ...
Spine Feast's user avatar
  • 4,780
196 votes
21 answers

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 ...
crasic's user avatar
  • 4,879
119 votes
21 answers

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, ...
NotSuper's user avatar
  • 1,863
91 votes
4 answers

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 ...
bouma's user avatar
  • 1,135
1 vote
2 answers

Inclusion–exclusion principle; what is $(-1)^{n+1}$

could somebody kindly confirm that my understanding of inclusion-exclusion matches it's formula. for a 3 sets example; we add 3 unions, subtract the total of all 3 pairwise intersections and add the ...
ManOnTheMoon's user avatar
34 votes
12 answers

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= ${{n}...
user avatar
5 votes
2 answers

Applied Probability- Bayes theorem

I need help in all things related to identifying, defining conditions and solution feed back and reasoning most importantly. 1) A blood test indicates the presence of a particular disease 95 % of the ...
user418682's user avatar
84 votes
13 answers

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 ...
Avicinnian's user avatar
44 votes
14 answers

If you draw two cards, what is the probability that the second card is a queen?

We had this question arise in class today and I still don't understand the answer given. We were to assume that drawing cards are independent events. We were asked what the probability that the second ...
Ampage Green's user avatar
71 votes
5 answers

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 $m$, ...
Sasha's user avatar
  • 70.7k
24 votes
1 answer

Gamma Distribution out of sum of exponential random variables

I have a sequence $T_1,T_2,\ldots$ of independent exponential random variables with paramter $\lambda$. I take the sum $S=\sum_{i=1}^n T_i$ and now I would like to calculate the probability density ...
TI Jones's user avatar
  • 531
65 votes
3 answers

product distribution of two uniform distribution, what about 3 or more

Say $X_1, X_2, \ldots, X_n$ are independent and identically distributed uniform random variables on the interval $(0,1)$. What is the product distribution of two of such random variables, e.g., $Z_2 =...
lulu's user avatar
  • 1,008
45 votes
4 answers

Pdf of the difference of two exponentially distributed random variables

Suppose we have two independent random variables $Y$ and $X$, both being exponentially distributed with respective parameters $\mu$ and $\lambda$. How can we calculate the pdf of $Y-X$?
Mathematics Lover's user avatar
182 votes
7 answers

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?
jamaicanworm's user avatar
  • 4,524
84 votes
8 answers

Choose a random number between $0$ and $1$ and record its value. Keep doing it until the sum of the numbers exceeds $1$. How many tries do we need?

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 ...
user25329's user avatar
  • 1,047
40 votes
3 answers

Expectation of the maximum of i.i.d. 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 .. n}X_n\...
Rawling's user avatar
  • 922
115 votes
14 answers

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?
leava_sinus's user avatar
  • 1,263
112 votes
3 answers

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? ...
Chris Taylor's user avatar
7 votes
1 answer

Why is flipping a head then a tail a different outcome than flipping a tail then a head?

In either case, one coin flip resulted in a head and the other resulted in a tail. Why is {H,T} a different outcome than {T,H}? Is this simply how we've defined an "outcome" in probability? My main ...
James Ronald's user avatar
  • 2,331
18 votes
3 answers

How exactly are the beta and gamma distributions related?

According to Wikipedia, the Beta distribution is related to the gamma distribution by the following relation: $$\lim_{n\to\infty}n B(k, n) = \Gamma(k, 1)$$ Can you point me to a derivation of this ...
Sten Linnarsson's user avatar
44 votes
2 answers

Probability that two random numbers are coprime is $\frac{6}{\pi^2}$

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 ...
uforoboa's user avatar
  • 7,094
13 votes
2 answers

$P(X>0,Y>0)$ for a bivariate normal distribution with correlation $\rho$

$X$ and $Y$ have a bivariate normal distribution with $\rho$ as covariance. $X$ and $Y$ are standard normal variables. I showed that $X$ and $Z= \dfrac{Y-\rho X}{\sqrt{1-\rho^2}}$ are independent ...
user669083's user avatar
  • 1,111
132 votes
8 answers

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 ...
Adnan Ali's user avatar
  • 1,493
2 votes
3 answers

Describing Bayesian Probability

I'm a CS major doing some work with image recognition in which I use Bayesian probability. I have to give a presentation on my work, and while I have no problem describing the CS portion, I'm less ...
Ryan's user avatar
  • 1,651
37 votes
7 answers

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 ...
Jozef's user avatar
  • 7,110
10 votes
3 answers

Find the Mean for Non-Negative Integer-Valued Random Variable

Let $X$ be a non-negative integer-valued random variable with finite mean. Show that $$E(X)=\sum^\infty_{n=0}P(X>n)$$ This is the hint from my lecturer. "Start with the definition $E(X)=\sum^\...
karfai's user avatar
  • 494
36 votes
6 answers

Probability that a quadratic polynomial with random coefficients has real roots

The following is a homework question for which I am asking guidance. Let $A$, $B$, $C$ be independent random variables uniformly distributed between $(0,1)$. What is the probability that the ...
Pedro d'Aquino's user avatar
25 votes
1 answer

Uniform distribution on the surface of unit sphere

It is known that given $X=(X_1, X_2, \ldots, X_n)$ iid $\sim N(0,1)$, then $X/\sqrt{X_1^2+\cdots+X_n^2}$ is uniformly distributed on the surface of unit sphere. Intuitively, I know that that's ...
Julie's user avatar
  • 1,107
21 votes
2 answers

Looking for examples of Discrete / Continuous complementary approaches

Among many fascinating sides of mathematics, there is one that I praise, especially for didactic purposes : the parallels that can be drawn between some "Continuous" and "Discrete"...
Jean Marie's user avatar
  • 82.9k
82 votes
7 answers

Poisson Distribution of sum of two random independent variables $X$, $Y$

$X \sim \mathcal{P}( \lambda) $ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim \mathcal{...
user avatar
60 votes
6 answers

Why does the median minimize $E(|X-c|)$?

Suppose $X$ is a real-valued random variable and let $P_X$ denote the distribution of $X$. Then $$ E(|X-c|) = \int_\mathbb{R} |x-c| dP_X(x). $$ The medians of $X$ are defined as any number $m \in \...
Tim's user avatar
  • 47.5k
76 votes
9 answers

Probability of 3 people in a room of 30 having the same birthday

I have been looking at the 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. ...
irl_irl's user avatar
  • 863
168 votes
13 answers

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.
21 votes
5 answers

Probability of tossing a fair coin with at least $k$ consecutive heads

Tossing a fair coin for $N$ times and we get a result series as $HTHTHHTT\dots~$, Here '$H$' denotes 'head' and '$T$' denotes 'tail' for a specific tossing each time. What is the probability that ...
Benson's user avatar
  • 519
12 votes
4 answers

Hitting probability of biased random walk on the integer line

Lets say we start at point 1. Each successive point you have a, say, 2/3 chance of increasing your position by 1 and a 1/3 chance of decreasing your position by 1. The walk ends when you reach 0. ...
A.S's user avatar
  • 9,016
17 votes
6 answers

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?
user avatar
15 votes
3 answers

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 $Z=X_1/X_2$?
Fan Zhang's user avatar
  • 1,967
67 votes
11 answers

Average Distance Between Random Points on a Line Segment

Suppose I have a line segment of length $L$. I now select two points at random along the segment. What is the expected value of the distance between the two points, and why?
Kenshin's user avatar
  • 2,160
25 votes
3 answers

Probability, conditional on a zero probability event

Is there a way to resolve probability of an event, given another event that never happens? Mathematically speaking the problem is: Given that $P(B) = 0$, $$P(A|B)=\frac{P(A \cap B)}{P(B)} = \frac{0}{...
Phonon's user avatar
  • 4,028
166 votes
15 answers

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, ...
Shaurya Gupta's user avatar
48 votes
6 answers

Probability that n points on a circle are in one semicircle

Choose n points randomly from a circle, how to calculate the probability that all the points are in one semicircle? Any hint is appreciated.
NECing's user avatar
  • 4,105
47 votes
6 answers

Expected Value of a Binomial distribution?

If $\mathrm P(X=k)=\binom nkp^k(1-p)^{n-k}$ for a binomial distribution, then from the definition of the expected value $$\mathrm E(X) = \sum^n_{k=0}k\mathrm P(X=k)=\sum^n_{k=0}k\binom nkp^k(1-p)^{n-k}...
user avatar
16 votes
4 answers

Why is the probability that a continuous random variable takes a specific value zero?

My understanding is that a random variable is actually a function $X: \Omega \to T$, where $\Omega$ is the sample space of some random experiment and $T$ is the set from which the possible values of ...
davitenio's user avatar
  • 1,236
10 votes
3 answers

CDF of absolute value of difference in random variables

Let $X$ and $Y$ be independent random variables, uniformly distributed in the interval $[0,1]$. Find the CDF and the PDF of $|X - Y|$? Attempt Let $Z = |X - Y|$, so for $z \geq 0$, the CDF $F_{Z}(z) =...
buzaku's user avatar
  • 599
26 votes
2 answers

Summing (0,1) uniform random variables up to 1 [duplicate]

Possible Duplicate: 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? So I'm reading a book about simulation, ...
Haile's user avatar
  • 477
16 votes
4 answers

Tail sum for expectation

In Pitman's Probability, the tail sum formula for expectation is introduced for a nonnegative (0,1,...) discrete random variable $X$: $$E(X) = \sum_{i=0}^\infty P(X > i).$$ I wonder if there is a ...
Tim's user avatar
  • 47.5k
10 votes
4 answers

Conditional expectation for a sum of iid random variables: $E(\xi\mid\xi+\eta)=E(\eta\mid\xi+\eta)=\frac{\xi+\eta}{2}$

I don't really know how to start proving this question. Let $\xi$ and $\eta$ be independent, identically distributed random variables with $E(|\xi|)$ finite. Show that $E(\xi\mid\xi+\eta)=E(\eta\mid\...
kkk's user avatar
  • 171
41 votes
5 answers

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 $\max(X,Y)$?
John Smith's user avatar

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