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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-4
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0answers
34 views

Checking if a coin is fair [on hold]

I want to check if a coin is fair (lands 50% of the times on each side). I flip that coin multiple times and count the number of times it fell on heads and the number of times it fell on tails and ...
1
vote
2answers
52 views

Flipping coins- percentages of heads vs tails [on hold]

If I flip a coin multiple times and count the number of time it fell on heads and the number of times it fell on tails and keep a track of them. In how many flips on average will the delta between ...
0
votes
0answers
9 views

Existence of Joint Distribution from Overlapping Marginal Distribution

Suppose $x_i\in \mathbb{R}^{n_i}$ for $i=0,1,...,k$. For each $i=1,...,k$, suppose $F_i$ is a probability measure of $(x_0,x_i)$ on $\mathbb{R}^{n_0 + n_i}$. Assume all $F_i$ have the same marginal ...
0
votes
2answers
31 views

why probability is multiplied in finding out dependent probabilities?

Why is probability multiplied in case of dependent events? When we want to find out say, We take out a card from a deck of 52 cards and take another without replacing, We get the probability of ...
2
votes
0answers
14 views

sampling requirements in probabilistic polynomial identity testing

In the Schwartz–Zippel algorithm for bounded error probabilistic polynomial identity testing, the main theorem is the following: For a non-zero multivariate polynomial $p(x_1,...,x_n)$ of total ...
0
votes
2answers
20 views

Card Probability Without Replacement

So there are 15 cards total, 5 red, 7 orange, and 3 yellow. At random you pick 3 (no replacement). What's the probability of picking: 1) Exactly 2 Red? 2) Not more than one yellow? 3) One of each? ...
0
votes
2answers
30 views

Average Goals Per Game

Okay I am trying to work something out. If for example Team A scored an average of 2.84 goals per game over a period of 95 matches. What is the probability that there next match will be 3 goals or ...
3
votes
0answers
25 views

Suggestions for Constructing a Random Variables from Correlated Observations

Let $\mathcal{X} \neq \phi $ be a finite set. Let $P_{XY_1Y_2}$ be a fixed joint distribution over $\mathcal{X}\times\mathcal{X}\times\mathcal{X}\ $ and that a random sample $(X,Y_1,Y_2 )$ is drawn ...
0
votes
0answers
11 views

Deriving global probabilities from local dynamics

I am interested in growth dynamics and, in particular, how to derive difference/differential/stochastic equations from local behavior of a system. For concreteness, let's imagine a simple predator/...
1
vote
1answer
32 views

Finding probability with the help of combinations

$N$ tutors are to be assigned to $s$ students with any student having at most one tutor and similarly any tutor having at most one student. If any tutor is assigned randomly then how can we find the ...
0
votes
2answers
27 views

Value of c so that $c(2-|x|-|y|)$ is a probability distribution function(see picture)

Hint: Use the formula of volume of pyaramid. My approach: I know that the integral of a pdf from $-\infty to +\infty$ gives you $1$. I tried taking the double integral, but got stuck in as how to ...
1
vote
1answer
39 views

Expected length of a random walk

Let $G = (V,E)$ be a connected graph. Now consider a random walk on $G$, where we pick a random vertex $v_0$ sampled uniformly at random from $V$. Let $v_i \in V$ denote the vertex in the current ...
1
vote
0answers
18 views

Finding Future probability of results within a group of numbers

I apologize in advance if this question, or one similar, has been asked - I couldn't find anything via a quick search. How do I go about finding the probability of any particular number from within ...
1
vote
1answer
29 views

Scale invariance of uniform distribution over $\mathbb R^2$?

If we make a uniform distribution of points over $\mathbb R^2$ with 1 point on average per unit square. And we zoom far out and make a density plot (give a color to each cell according to how many ...
0
votes
2answers
52 views

Intuition of the expectation.

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space. What is the intuition behind the definition $$\mathbb E[X]=\int_\Omega X(\omega) \, \mathrm d \mathbb P(\omega )\text{ ?}$$ I don't see in ...
2
votes
2answers
65 views

If you roll two six-sided dice, what is the probability that the dice add to 10 or higher?

When answering these sort of questions people mostly resort to diagrams and I'm wondering if there is a way to calculate the probability without going through each outcome, just solely on the given ...
2
votes
1answer
22 views

Understanding the Skorohod-space

I am having a lack of understanding the Skorhodspace considering cadlag processes. A random variable $X$ is measurable mapping between two measure spaces say $(\Omega,\mathcal{F})\mapsto (\tilde{\...
1
vote
0answers
27 views

PDF/CDF of max-min type random variable

For i.i.d. random variables, we may write the CDF of $t=\max(t_1,\cdots,t_N)$ as $$F_t(t)=F_{t_i}(x)^n$$ and the CDF of $x=\min(x_1,\cdots,x_N)$ as $$F_x(x)=1-(1-F_{x_i}(x))^n$$ When we have $X=\...
1
vote
1answer
20 views

Expected Value when joint density function is given

Let $X_i$ denote the percentage of votes cast in a given election that are for candidate $i$, and suppose $X_1$ and $X_2$ have a joint density function $f_{X_1,X_2}(x,y)= 3(x+y)$ if $x\geq0;y\geq0;0\...
1
vote
2answers
53 views

Probability Conjecture

I think there is a flaw in my logic but I'm not sure where it would be. Let HHH denote the event of three coin flips. Let E(HHH) be the expected value of the number of coin flips until HHH. Let E(...
4
votes
1answer
32 views

probability/combinatorics question with marbles

An urn has 20 green out of 50 marbles. Draw all 50 marbles without replacement. Let X = # of green marble runs of any length. Example : GGGGBBBGGBBGBB. . . In the above example, there are 3 runs in ...
0
votes
2answers
36 views

Calculating the expectation of binomial distribution without calculating the summation

We know that expectation of a binomial distribution is $$\sum _{1}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){p}^{k}{\left(1-p\right)}^{n-k}k = np$$ But while proving it, it is being written ...
1
vote
1answer
22 views

Basic query Related to dependent random variables

$X$ and $Y$ are two dependent random variables. I want to find the following probability $$\Pr(2X<c,4Y>c)$$ wher $c$ is some positive number. In my attempt, I can expand the above probability ...
-2
votes
2answers
31 views

measure of a set which is a subset of infinitely many subsets of probability measure space [on hold]

Let $B,A_1,A_2,....$ be the subsets of a probability measure space. If $ B \subset \bigcup A_j$, show that $m(B) \le \sum_{j=0}^\infty m(A_j)$. I have no idea as how to approach it. I do have the ...
4
votes
2answers
110 views

Expected number of virus cells

I've found this question in a past programming assignment from a course I'm currently reading. Its statement looks like this : A recent lab accident resulted in the creation of an extremely ...
0
votes
1answer
54 views

Poisson distribution with more than one lambda.

An archaeologist has two old pieces of wood and shall decide which piece of wood is the oldest. Radioactivity from the pieces of wood are recorded by a counter. Number of registrations per unit time ...
-2
votes
1answer
45 views

Loaded coin: probability I will never get two heads [on hold]

I'm currently starting to learn probability in an intro course in college and am wondering how to solve this. Given a loaded coin that gives a 60% chance of flipping heads, and 40% chance of tails, ...
1
vote
1answer
53 views

Is this betting game profitable?

I'm wondering whether a specific betting game is profitable but I'm not quite sure how to analyse it, some good tips on how to start would be great. Suppose a fair coin is tossed repeatedly. ...
0
votes
0answers
26 views

Upper bound number of self-avoiding walk of length $n$

A self-avoiding walk is a sequence of $n$ neighbor sites on a graph which are all distinct. Let $C_n$ be number of self-avoiding walk of length $n$ in a $d$-dimensional regular lattice. As a ...
0
votes
0answers
36 views

A game of blackjack with guaranteed 50 50 win loss

Suppose you are playing 2 hands of blackjack at the same time and the dealer guarantees if you lose hand A they will pay whatever you bet for hand B but if you win hand A you automatically lose hand B....
0
votes
1answer
47 views

What is the expectation of $X$ given $XY > \alpha$

Suppose $X$ and $Y$ are non-negative independent random variables. Then how can I evaluate $\mathbb E[\ X\ |\ XY > \alpha\ ]$ for some $\alpha > 0$? If I know the precise distributions of both ...
2
votes
0answers
26 views

Probability calculation on 2 similar vases with difference in time and backlog

Below is a problem that we find hard to solve, any input would be much appreciated. In short, we're looking for the exact probability after n tries, and the corresponding formula for this problem. ...
-1
votes
1answer
42 views

Loaded die problem.

A die is loaded in such a way that the probability that a 6 is thrown is five times that of any other number, each of them being equally probable. What is the ratio of the probability of obtaining a ...
1
vote
1answer
32 views

Number of $n$-ples of integers summing to a given integer

Fix a non-negative integer $m$. For any integer $A \geq 1$, use $P_m(A)$ to denote the number of ways of rewriting $A = A_1 + A_2 + \ldots A_m$, with $A_i$ non-negative integers (eventually $0$). Is ...
1
vote
0answers
13 views

Probability of Hamming distance being close to accurate

Consider two $n$-dimensional integer valued vector $v,w$ with $v_i,w_i \in \{1,\dots, n\}$. Now consider a random function $f:\{1,\dots,n\} \rightarrow \{1,\dots,1/\epsilon\}$ with integer $0< \...
0
votes
1answer
38 views

Number of ways of writing an integer as a sum of other integers

Fix a non-negative integer $m$. For any integer $A \geq 1$, use $P_m(A)$ to denote the number of ways of rewriting $A = A_1 + A_2 + \ldots A_k$, with $ 1 \leq A_i \leq m$, for any $k \geq 1$. I ...
0
votes
3answers
129 views

Integral that makes square root of $\frac{\pi}{2}$ [duplicate]

My question is regarding a integral that´s giving me a huge headache. I want to show $$\int_{0}^{\infty}y^2e^{-\frac{y^2}{2}}dy=\sqrt{\frac{\pi}{2}}$$ I'm studying for an exam. I'm suppose to find ...
-1
votes
0answers
15 views

Binomial distribution: dice [on hold]

A fair die is rolled five times. Find the probability of obtaining: a) all results greater than 3 b) a 5 on the first roll only c) a 5 on the second and third roll only
2
votes
4answers
67 views

Probability that the second throw of a fair die exceeds the first

A player throws an ordinary die and records the score $A$. The player then throws the die again and again records the score, $B$. if $B>A$ then we set a score for this player. What is the ...
0
votes
0answers
18 views

Difference between a mixture of distributions and a convolution. Intuition

From what I could gather Mixture: if $X_i\sim^{iid} f_i$, then W is a mixture with $f_W =\sum \frac{f_i}{n}$. This definition could also be for the CDF instead of the density. Convolution: To make ...
0
votes
0answers
12 views

Help needed related to derivation

I want to find the following probability $$P(z_i\leq min(1,x^{-m})<z_{i+1}, x<x_1|z_i \leq 1, z_{i+1}>1)$$ where $m$ is some value greater than $2$, $z_i$'s are some constants and pdf of $x$ ...
1
vote
1answer
26 views

Probability of being in a circle, given normal

Let's assume a bivariate normal distribution with center $\mu$ and covariance matrix $\Sigma$. Let a circle $C$ be given as $C=\{x\in\mathbb{R}^2:||x-\mu||\leq R\}$. I would like to calculate the ...
0
votes
2answers
27 views

Probability - Poisson Arrival Process

Car arrive at a toll booth according to the Poisson process at a rate of 3 arrivals per minute. a) What is the probability that the third car arrives within 3 minutes of the first car? b) Of the ...
0
votes
1answer
20 views

Is this equality holds? $\overline{F^{*2}}(x)=\int_0^x\overline{F}(x-y)dF(y)$

$X_1,X_2$ are non-negative i.i.d random variables with CDF F(x). I have a problem proving that following identity holds. $$ \frac{\overline{F^{*2}}(x)}{\overline{F}(x)}=1+\int_0^x\frac{\overline{F}(...
1
vote
3answers
60 views

I am stucked in some of this question about sample space and probability

What is the size of the sample space for the following scenario: Roll $3$ six-sided dice, and discard the highest roll In this question I know that sample space is all possible outcomes so the ...
0
votes
3answers
65 views

Probability of rolling 1-8 using six-sided dice

If I roll two six-sided dice where the first die is valued simply 1-6, but the second die is valued as 1-2=0, 3-4=1, and 5-6=2, and I total the two dice, will the probability of the numbers 1-8 be ...
1
vote
2answers
26 views

Probability of selecting same factor.

Willie Pikette randomly selects a factor of $144$. Betty Wheel selects a factor of $88$. What is the probability that they selected the same number? This is my incorrect approach (and please feel ...
2
votes
1answer
47 views

Conditional expectation of a product XY given Z with Y independent of Z

Let $X,Y$ and $Z$ be integrable random variables s.t. $XY$ is integrable and $Y$ is independent of $Z$ . I was wondering if there are any helpful/common ways of rewriting $\mathbb{E}[XY\mid ...
1
vote
0answers
24 views

Problems on continous random variable, probability, estimates.

Problem image: answering to questions (i) and (ii) i found that: $$pdf: f(x) = x \frac{2}{\theta^2}$$ $$E(x) = \frac{2}{3} \theta$$ $$Var(x) = \frac{1}{18}\theta^2$$ And now I tried to answer ...
0
votes
1answer
31 views

For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely

Question in the title: For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely My main problem is that I don't even understand what $E (Xh(Y)|Y)$ means.....