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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2
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0answers
26 views

Conditional expectation of the sum of two random variables

I've got some difficulties in calculating the conditional expectation of the sum of two RV. I am not sure if I correctly formalized the scenario I am looking at. So I am trying to describe it first: ...
2
votes
1answer
16 views

The probability of getting a certain image by random pixelation

Well, seeing that I'm terribly bad at math I don't know how to solve this, I'll try to explain, excuse me if I sound dumb. Just suppose that I've got a photo/image with 320x240 resolution and 24 bit ...
0
votes
1answer
35 views

Expected number of changes of serves in a game of raquetball

Suppose a game of racquetball is being played, with players A and B. Assume further that A starts the play first, that the winner of a point serves the next point, and that the match lasts until the ...
0
votes
1answer
37 views

Probability of two teams meet up in FIFA tounament

**Second round is single elimination round. **Tournament is from 16 teams elimination follower by quarter finals,semi-finals and a final. **The losing semi-finalist contest a third place match. In ...
-5
votes
0answers
31 views

what is the probability of getting two heads twice in 5 tosses of two coins? [on hold]

If I have two identical fair coins and toss them both at the same time, what is the probability that after five tosses, two tosses resulted in both coins landing heads up?
0
votes
2answers
25 views

probability of a flipped coin

A fair coin is flipped three times. Let $A$ be the event that a head occurs in the first flip and $B$ be the event that exactly one head occurs. a) Find $p(A/B)$ b) Are $A$ and $B$ independent? ...
2
votes
1answer
31 views

Expectation related to Normal distribution and its density

Given $\sigma^2>0$. Let $Z\sim N(0,1)$ and $\Phi$ be the cumulative standard normal with density function $\phi$. I wish to show that $$ E\left(\frac{Z^2}{[\phi(\sigma Z)]^2}\Phi(\sigma ...
-1
votes
2answers
30 views

If pages in a book have an iid Poisson number of errors, in 10 pages what is the probability that exactly 3 pages have exactly 1 error?

Suppose the number of spelling error on any given page in particular book can be modeled by a Poisson distribution with $\lambda=2$, and assume that the number of errors on different pages is ...
-1
votes
0answers
37 views

Choosing random marbles until one is divisible by $X$ [on hold]

A box contains twelve marbles on which they are numbered by $1,2,3,...,12$. Now let $X$ represent the number of marbles you must choose with replacement until you obtain one with a number that is ...
1
vote
1answer
38 views

Probability of getting a right answer?

A student is taking a $4$ question multiple choice quiz with each question having $5$ options. What is the probability that he will get at least one question correct? P.S. Please keep answers at ...
0
votes
2answers
41 views

Expectation of CDF of continuous random variable $X$, evaluated at $X$

Given the continuous random variable $X$ with cumulative distribution function $F_{X}$, find $E[F_{X}(X)]$. Attempt at solution: I understand that the expected value, $E[X]$, of a random ...
0
votes
2answers
26 views

Convergence of running maximum of uniform random variables [on hold]

Let $X_1, X_2, ... X_n$ be an IID sequence of IID random variables that have a uniform distribution $(0,1)$. Let Max$(n) =$ max$(X_k:1\le k \le n)$, where $n\in \mathbb N$. How do I show that ...
1
vote
2answers
36 views

Confusion regarding the fixed point $p(x) =x$

Consider an empty urn. Now at each time, we are adding one ball to it, Either red or black, the probability of a red ball being added depends on $x$ ($x$ denotes the current fraction of red balls in ...
0
votes
2answers
48 views

Expected value and variance of max{x, y}

I've run into this problem while playing a game called Europa Universalis 4. I've done similar maths before in my studies so I'm pretty sure this should have an easy answer but I can't for the life of ...
1
vote
1answer
45 views

Calculate single “battle” outcome odds for RISK

I am trying to reproduce the values in this odds ratio table from Wikipedia. For all those unfamiliar with RISK, this is a game where units fight against each other via the roll of the dice: The ...
1
vote
0answers
20 views

change some element of a correlation matrix

I am working on correlation matrices. These matrices have the main property to be symetric , positive-semidefinite, have 1 on the diagonal and each of their elements is between -1 and 1. Let's say I ...
0
votes
0answers
14 views

Stopped strong Markov process again strong Markov?

Following setting: I have a right-continuous strong Markov process X in a right-continuous filtration >$\mathbb{F}=(F_t)$ and a P-a.s. finite stopping time $\tau$. My question is: Is the ...
1
vote
1answer
15 views

Convex and Concave Functions using Known Function Values

I am reading the classic Prospect Theory: An Analysis of Decision Under Risk (1979, Econometrica) by Kahneman and Tversky. I am not clear on something on page 278: ...
2
votes
2answers
41 views

Estimate the number of ants in a colony

A friend of mine gave me this weird problem I cannot solve. To estimate the number of ants in a colony an entomologist draws 5500 ants randomly from the colony, labels them with a radioactive isotope ...
26
votes
3answers
3k 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 ...
3
votes
3answers
55 views

Probability of a year which is not a leap year

If a 4 digit year is choosen randomly, what is the probability that it is not a leap year ? This problem has come in my exam and i have written like this I know that the number of four digit year ...
1
vote
2answers
35 views

what is the distribution of a uniform r.v. divided by an exponential r.v.?

Say $U=\frac{X}{Y}$. X and Y are independent with each other. X is a Uniform distribution r.v. $X\sim \mathcal{U}(0,1)$. Y is an exponential distribution r.v., $Y\sim\mathcal{Exp}(\lambda)$, whose pdf ...
0
votes
1answer
56 views

Probability and coin tosses

Taking a Probability & Statistics class this term and trying to get my head wrapped around how I calculate coin tosses with specific out comes in mind. We're using the nCr and nPr functions on our ...
1
vote
3answers
32 views

Determine the law of $F^{-1}(U)$, $U$ uniformly distributed on $[0,1]$

i'm trying to understand the following problem Let $X$ be a real random variable, its distribution function is $F(t):\Bbb{P}(X\le t), \forall t\in \Bbb{R}$. Define the right-continuous inverse by ...
2
votes
2answers
50 views

conditional probability about sum and product rule

I am reading Bishop's Pattern Recognition and Machine Learning. In page 73, chapter 2.1. I can't understand the formula 2.19 : $$p(x=1|\mathcal{D})=\int_0^1 p(x=1|\mu)p(\mu|\mathcal{D})\text{d}\mu ...
0
votes
0answers
23 views

What is the optimal prize for a prize ticket in a raffle [on hold]

What, if any is the optimal price for a prize ticket given the value of a prize? For example if you were to raffle a TV and wanted to cover the cost of the prize? Let say the people were aware of how ...
0
votes
2answers
27 views

conditional probability maybe?

If in application A, 70% of the users are men and 30% women. In application B, 60% men and 40% women. Given you have both applications, what is the probability that you are a man?
-2
votes
1answer
28 views

Baye's theorem may be required. [on hold]

A message is sent which consists of $n$ binary symbols $0$ and $1$. Each symbol is distorted with a small probability $p$ (is changed to the opposite). To be on the safe side the message is repeated ...
0
votes
0answers
25 views

Kelly criterion for 3 outcomes

I have been exploring the Kelly criterion for optimizing the bet size for a two outcome bet situation. I'm having trouble applying this to a three outcome bet. I may refer to this excellent thread: ...
-2
votes
1answer
24 views

Expected Value Question Intermediate [on hold]

Mila has four ropes. She chooses two of the eight loose ends at random (possibly from the same rope) and ties them together, leaving six loose ends. She again chooses two of these six ends at random ...
-2
votes
1answer
24 views

Binomial Probability help [on hold]

The problem is: $35\%$ percent of the employees in a company receive an incentive in the month of April. What is the probability that $4$ employees of the company chosen at random do not receive the ...
1
vote
1answer
30 views

Sum of truncated normal random variable and normal random variable

I'm wondering if there is a closed-form pdf of sum of "correlated" normal random variable and truncated normal random variable. I found a paper providing the pdf for "uncorrelated" case, but could ...
0
votes
2answers
42 views

Check if a given function is a probability density function [on hold]

Given $f(x)=\tfrac1{π(1+x^2)}, ~x\in(-\infty, \infty)$, is it true that $f$ is the probability density function of some continuous random variable?
2
votes
1answer
53 views

Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
2
votes
1answer
37 views

A question about Malliavin calculus

An application of Malliavin calculus is to calculate the sensitivity of financial Greeks. However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to ...
1
vote
1answer
36 views

Kids and cookies, probability

10 children $D_1,D_2,...,D_{10}$ were given 20 cookies. What's the probability that $D_{10}$ has at least one cookie if we know that $D_1$ and $D_2$ both have exactly 2 cookies. I think that by ...
0
votes
1answer
49 views

Fast way to inverse B'CB+D

$\mathbf {A = B'CB}$, where $\mathbf A$ is of dimension $n \times n$, $\mathbf C$ is m by m, positive definite and symmetric, $\mathbf B$ is of dimension $m \times n$, and $n >> m$. Inversion ...
3
votes
1answer
42 views

Statistical test for “too perfect” random number generator?

I am attempting to characterize some random number generator programs in a very simple way. Specifically, I'm rolling a simulated 6-sided die $3 \times 10^8$ times and keeping a count of how many ...
0
votes
1answer
22 views

Polynomial joint pdf $f(x,y)$ such that of $f(x) \neq f(y)$

How can I build a polynomial joint pdf $f(x,y)$ for $x \in [x_1, x_2]$ and $y \in [y_1, y_2]$ such that of $f(x) \neq f(y)$ or equivalently, $x$ and $y$ are depended on each other?
0
votes
1answer
25 views

conditional probability of throwing a dice

I would like to compute the conditional probability of throwing a dice. The event $A$ is getting 2 and the event $B$ is the number to be even, so the question is what is the probability of getting 2 ...
4
votes
4answers
122 views

What is the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$ and some of the properties of $E[X \mid Y]$?

I was trying to understand both intuitively and rigorously what the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$. Let me tell you first the things that do make sense to me. $E[X\mid Y=y]$ makes ...
1
vote
1answer
21 views

Probability of multiple variables, geometric distribution?

You are on a basketball team, and at the end of every practice, you shoot half-court shots until you make one. Once you make a shot, you go home. Each half-court shot, independent of all other shots, ...
0
votes
0answers
35 views

The mathematical odds of winning a hand in poker with two boards

In Hold-em, after the flop, one hand has two pairs and the other hand has a flush draw. The odds of two pairs winning against a flush draw after the flop with 2 cards to come is roughly 3:2. In ...
1
vote
1answer
36 views

Intersection of countable many sets of measure $1$

Consider a probability space $(X,\mathscr M,\mu)$ and a collection of measurable sets $\{A_n\}_{n\in\mathbb N}$ such that $\mu (A_n)=1$ for every $n$. Then I don't unterstand the following result: ...
6
votes
1answer
68 views

Probability that two circles in space are linked

Let $C_0$ be a circle centered on the origin, and $C_1$ a circle centered on $(1,0,0)$, center distance of $1$. Q1. If both $C_0$ and $C_1$ are randomly oriented and have the same radius $r ...
-1
votes
3answers
68 views

The chance to double 1000 points into 2000 points [on hold]

You own 1000 points. Your goal is to reach 2000 points, the only way you gain points is by gambling. You will always gamble 40 points, your chance of winning a 40 points gamble is 60%, how high is ...
0
votes
1answer
24 views

How do you get the probability distribution of the sum of random variables by using the inverse of the transform?

I read the following statement: If X and Y are independent random variables, the distribution of their sum W = X + Y can be obtained by computing and then inverting the transform $M_W (s) = ...
1
vote
2answers
56 views

Coin flip gamble

You have an amount of money to bet on a fair coin flipping and landing on heads. How much should you bet as a function of your balance to maximize your probability of profiting if you play $x$ times?
1
vote
2answers
48 views

let $X$ be a standard Gaussian random variable. Show that $(X,X)$ is not absolutely continuous.

i'm trying to understand a proof of the following statement: let $X$ be a standard Gaussian random variable. Show that $(X,X)$ is not absolutely continuous. I'll write down the proof in such a ...
4
votes
2answers
157 views

Drunken sailor's Random Walking

A drunken walker is on $x=0$, and if $x<0$, he falls and he dies.(Once he gets position $x<0$, he dies permanently.) There is $0<p<1$ chance to move right ($x \rightarrow x+1$), and $1-p$ ...