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.

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97
votes
3answers
82k 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 ...
41
votes
3answers
42k views

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 ...
25
votes
12answers
22k 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= ${{n}...
90
votes
20answers
197k 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, ...
125
votes
14answers
69k 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 ...
63
votes
4answers
38k 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 ...
72
votes
12answers
19k 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 ...
56
votes
5answers
19k 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 $m$, ...
20
votes
1answer
8k views

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 ...
44
votes
3answers
37k views

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 =...
102
votes
5answers
150k 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?
87
votes
2answers
50k 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? ...
18
votes
1answer
56k views

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 ...
34
votes
3answers
51k 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 $v-u$?
66
votes
12answers
95k views

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?
14
votes
2answers
7k 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 $Z=X_1/X_2$?
27
votes
3answers
15k views

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\...
31
votes
7answers
5k 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 ...
4
votes
2answers
476 views

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 ...
12
votes
4answers
8k views

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 ...
5
votes
3answers
6k views

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^\...
46
votes
7answers
122k views

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{...
13
votes
6answers
11k 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?
45
votes
10answers
41k views

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?
26
votes
2answers
8k views

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 ...
14
votes
6answers
2k views

Book on combinatorial identities

Do you know any good book that deals extensively with identities obtained using combinatorial and/or probabilistic arguments (e.g., by solving the same combinatorial or probability problem in two ...
64
votes
9answers
72k 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. ...
68
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 ...
27
votes
9answers
9k 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 ...
23
votes
1answer
6k views

Probability that a random binary matrix is invertible? [duplicate]

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 ...
294
votes
6answers
86k 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? ...
42
votes
15answers
77k views

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 ...
23
votes
4answers
17k views

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 ...
12
votes
2answers
509 views

Looking for examples of Discrete / Continuous complementary approaches

Among many fascinating sides of mathematics, there is one that I praise, especially in teaching at the boundary between undergraduate/graduate level : the parallels that can be drawn between a "...
5
votes
3answers
8k views

X,Y are independent exponentially distributed then what is the distribution of X/(X+Y)

Been crushing my head with this exercise. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything together. The exercise goes as ...
3
votes
2answers
1k views

What are numerical methods of evaluating $P(1 < Z \leq 2)$ for standard normal Z?

Let $Z \sim Norm(0, 1)$ and denote its PDF and CDF by $\phi$ and $\Phi$ respectively. Then, theoretically, $P(1 < Z \leq 2) = \Phi(2) - \Phi(1).$ However $\Phi$ cannot be expressed in closed form, ...
6
votes
3answers
576 views

Given that $X,Y$ are independent $N(0,1)$ , show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$ are independent $N(0,\frac{1}{4})$

It is given that $X,Y \overset{\text{i.i.d.}}{\sim} N(0,1)$ Show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}} \overset{\text{i.i.d.}}{\sim} N(0,\frac{1}{4})$ I was thinking of ...
259
votes
13answers
36k 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 ...
18
votes
6answers
24k views

Average length of the longest segment

This post is related to a previous SE post If a 1 meter rope …. concerning average length of a smallest segment. A rope of 1m is divided into three pieces by two random points. Find the average ...
21
votes
9answers
4k views

Intuition behind the Definition of Conditional Probability (for 2 Events)

What is some intuitive insight regarding the conditional probability definition: $P(A\mid B) = \large \frac{P(A \cap B)}{P(B)}$ ? I am looking for an intuitive motivation. My textbook merely gives a ...
21
votes
3answers
2k 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 suggests ...
38
votes
8answers
50k 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?
35
votes
6answers
51k views

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}...
28
votes
2answers
11k 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 $\frac14$ (coordinatize the stick form $...
28
votes
4answers
56k 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 $\max(X,Y)$?
8
votes
3answers
10k views

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. ...
17
votes
2answers
2k 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
4answers
19k views

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 ...
2
votes
2answers
4k views

$P(X>0,Y>0)$ 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 ...
2
votes
2answers
2k views

Coupon Collector Problem with Batched Selections

I am trying to solve a variation on the coupon collector's problem. In this scenario, someone is selecting coupons at random with replacement from n different possible coupons. However, the person is ...