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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2answers
46 views

Probability of $\max_i \{X_i\} = X_0$ where $X_i$ are iid binomial

We have $M$ Binomial random variables, where $X_0 \sim $ Bin$(n,p)$ and $X_i \sim $ Bin$(n,1/2)$. Suppose $p > 1/2$. I'm interested in the probability that $\mathbb{P}(\max \{X_1,\dots,X_M\} \geq ...
2
votes
5answers
87 views

Compare $\mathbb{E}[XY]\mathbb{E}[XY]$ with $\mathbb{E}[X]\mathbb{E}[XY^2]$

$\newcommand{\E}{\mathbb{E}}$So this was a question asked to me in an interview where $X$ and $Y$ are two random variables and I was asked to compare the $\E[XY]\E[XY]$ with $\E[X]\E[XY^2]$ . The ...
1
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2answers
33 views

How to designate the density function uniform distribution (continuous) on the set $[-1,0] \cup [3,5]$

How to designate the density function uniform distribution (continuous) on the set $[-1,0] \cup [3,5]$? I know how to do it when we only have for example [3,5] but for this set $[-1,0] \cup [3,5]$ I ...
1
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0answers
52 views

How to calculate $\mathbb{P}(A>B)$ using the Jeffrey Prior

Let say that you created 2 marketing campaigns. You sent 200 impressions on these campaigns as follow: Campaign A : Got 100 impressions and 2 successes with a value of 1.5$ per success Campaign B : ...
0
votes
0answers
49 views

Predicting certain independent events (each have a known probability)

Consider 15 independent events, each of these events can have three states (for example they can be 1, 2 or 3). Now for example in the event one there is 20% chance of getting 1, 45% of getting 2 ...
3
votes
1answer
55 views

How exactly is the St Petersburg Paradox giving bounded payoff in average-of-N-trials?

I understand why the expected value of the St Petersburg Paradox is algebraically infinite, but intuition tells me that in practice any given round of the game will not go on multiplying the pot for ...
0
votes
0answers
38 views

Central Limit Theorem for gambling return ratio

Consider a single bet with odds $o$ and thereby implied probability $1/o$. Assume that the real probability $p$ is known. Let $I$ be the stake, and $y$ the return from the bet. Then, $\mathbb{E}(y) ...
-1
votes
3answers
100 views

number of possible combinations of 5 from the set of numbers 1-10

I am trying to figure out how to set up this problem or any similar one. If you have 10 balls numbered from 1 to 10, and you pick 5 balls, what is the probability that you will have picked ball#1? In ...
0
votes
0answers
15 views

Queue depth to keep workers busy

I'm trying to find a probability of keeping w workers busy with a q queue depth feeding those w workers. When the queue has at least one item in it the item can be taken and the item was randomly ...
1
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1answer
46 views

What is the uncertainty of a discrete sum given the uncertainty of an individual element?

I have a measurement $$X=\sum_{i=1}^nX_i,$$ and I am interested to know standard deviation $\sigma_X^2$ of measurement $X$, assuming I know $\sigma_i^2$, the standard deviation of all measurements $...
-1
votes
0answers
37 views

Find out a probability from a set [closed]

I have set A = [ 1 0 0 1 0]; in this set, the number of 1's is two and the number of 0's is three. Question: How to calculate probability of randomly selecting a 1 from set A? (without replacement of ...
-1
votes
2answers
57 views

Probability problem

I created this problem based on the following probability riddle here. You're a king, and you were given two groups of people, and a certain information about them. First group has 2 people. One of ...
1
vote
3answers
90 views

How many days will it take me to earn a certain sum of money (given a certain probability)?

Suppose I want to earn $7000$. How many days will it take me to earn it, if there is an $80\%$ chance I will make $500$ on a particular day and a $20\%$ chance I will lose $1500$ on the same day? My ...
1
vote
1answer
47 views

What is the expected number of triangles contained in this graph?

I can't seem to understand this question and I really don't know where to start. Could someone please give an explanation as to how to go about answering this? A simple graph is formed randomly on ...
1
vote
2answers
34 views

Ticket lottery question

A hundred tickets are marked $1,2,3,...100$, and they are arranged at random. Four tickets are picked from these and given to four persons A,B,C,D. What is the probability that A gets the ticket ...
-1
votes
0answers
36 views

Probability: What is the condition for 3 random variables to be independent [closed]

I know that the condition for 3 events (A, B, C) to be independent is: $\mathbb{P}(ABC) = \mathbb{P}(A)\mathbb{P}(B)\mathbb{P}(C)$ But is it the condition the same for random variables X, Y, Z? For ...
1
vote
2answers
42 views

Logistic regression for football results - Estimating coefficient through maximum likelihood

Consider two football teams $V$ and $L$ with strengths $W_V$ and $W_L$, respectively. Let's assume that the draw probability $\mathbb{P}(Draw)$ is known. Then this model is supposed to give estimates ...
1
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0answers
33 views

Distribution of number of players who draw, with n independent games of chess

In a chess tournament, n games are being played, independently. Each game ends in a win for one player with probability 0.4 and ends in a draw (tie) with probability 0.6. Find the PMFs of the number ...
1
vote
1answer
37 views

Let $E$ := {$U_1 \geq U,U_2 \geq U,U_3 < U,U_4 \geq U, U_5 < U,U_6 \geq U,U_7 \geq U$}

Let $U,U_1,U_2,...$ be independant, on [0,1] uniform distributed random variables. Let $E$ := {$U_1 \geq U,U_2 \geq U,U_3 < U,U_4 \geq U, U_5 < U,U_6 \geq U,U_7 \geq U$}. Find the probabiliy $...
2
votes
1answer
26 views

Understanding the flat (uniform) Dirichlet distribution density over a simplex

This should be really straightforward from the formula, but somehow I'm having trouble understanding the density of a Dirichlet distribution with $\alpha = [1, 1, ... 1] \in R^k$, which is a uniform ...
1
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3answers
57 views

Probability: Are disjoint events independent? [duplicate]

I just read that disjoint events $\mathbb{P}(AB) = 0$ are independent. This really frustrates me. My teacher stated otherwise - $\mathbb{P}(AB) = 0 \iff A \cap B = \emptyset \implies \mathbb{P}(AB) =...
3
votes
1answer
42 views

How much area in a unit square is not covered by $k$ disjoint disks of maximal area centered at random points within the square?

1. Paint a $1\times 1$ square in blue. 2. Take $k$ points randomly and uniformly from the square. 3. Paint $k$ disks centered at each point in red. The radius of the disk centered at point $...
3
votes
1answer
73 views

Help required in finding solution to overdetermined system of equations?

I have access to M probability measures, $P_e(c_1),P_e(c_2),\cdots,P_e(c_M)$, defined as \begin{equation} P_e(x) = p(x|y) = p(y|x)\cdot \mathbb{P}(X=x) \frac{1}{\sqrt{2\pi\sigma^2}} \exp\Big[-\frac{(y-...
7
votes
4answers
146 views

Find the probability that a word with 15 letters (selected from P,T,I,N) does not contain TINT

If a word with 15 letters is formed at random using the letters P, T, I, N, find the probability that it does not contain the sequence TINT. (I just made up this problem.)
1
vote
1answer
33 views

Left continuous of a CDF.

I am working through Oksendal SDEs book and have a question about an exercise (number 2.2): $X: \Omega \rightarrow \mathbb{R}$ is a r.v with $F(x)= \mathbb{P}[X \leq x]$. We can show $F$ is ...
-2
votes
0answers
14 views

problem in concept of linear and nonlinear process [closed]

Is the nonlinear process is nonstationary process? in the other word: what is the relationship between stationary and linearity?
1
vote
1answer
23 views

Calculation of probability of event intersection

This is a question from MIT 6.041 open courseware. Most mornings, Victor checks the weather report before deciding whether to carry an umbrella. If the forecast is “rain,” the probability of ...
2
votes
2answers
60 views

How to find the probability of three friends out of five friends coming to pick you up?

I am working on a probability course with following problem: You are stranded by the road, so you decide to call some of your friends. You know they are not very trustworthy– in fact, each one ...
0
votes
0answers
67 views

Explanation of the proof of the expectation of a linear function.

$\newcommand{\var}{\operatorname{Var}}$ T1. LINEAR FUNCTIONS. For linear functions, the expectation of the function is the function of the expectation, and the variance of the function is the ...
1
vote
1answer
46 views

Notation for probability: $C_n^r$, $P_n^r$, $A_n^r$?

I was told that $C^{n}_{k}$ refers to combinations or choose k elements from n elements, $\bar{C^{n}_{k}}$ refers to combinations with repetitions (i.e. $C^{n+k-1}_{k}$), and $P^{n}_{k}$ refers to ...
1
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0answers
37 views

Approximating Geometric Brownian Motion numerically

I am trying to generate a numerical solution to the SDE for Geometric Brownian Motion. The stochastic process is given by $S_t = \exp(\sigma W_t + \mu t)$, and by Ito's lemma, we have that the SDE is ...
-1
votes
0answers
33 views

Bayesian posterior probability [closed]

Let's say that you are in a casino and you have played on 3 different slot machines following this flow: Slot machine A, play 10 times, win 2 times for a total of 2$ Slot machine B, play 100 times, ...
0
votes
3answers
82 views

Probability: Prove that events are independent

I'm self-learning probability and struggle on the following task: If $A$ and $B$ are independent events, prove that $A \cup B$ and $A \cap B$ are also independent. This is one of those cases where ...
1
vote
1answer
34 views

Conditional negative binomial

An urn contains white and black balls with $p_w=p$ and $p_b=1−p$. Some extractions with replacement are made. $X_a$ is the random variable representing the number of extractions made in order to get ...
2
votes
1answer
65 views

How does probability change the more times you perform a procedure?

I have a question based on fair coins. Every round, two coins are flipped. If both are heads, we say "Success" and end the experiment. What is the probability of saying "Success" in any round i? ...
6
votes
1answer
120 views

Picking pairs of socks from a drawer.

There are $n$ socks in a drawer, of $m$ different colours. Initially, the probability of picking a sock of colour $c_i$ at random is $\mathbb{P}(c_i) \cdot 2r$ socks are picked at random, without ...
1
vote
1answer
21 views

Multinomial distribution and conditional probability

An urn contains $w$ white, $b$ black and $r$ red balls. $n$ extractions with replacement are made. $X_w$, $X_b$ and $X_r$ are the random variables representing the number of white, black and red balls ...
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votes
0answers
57 views

Probability: How do I prove this inequality? [closed]

While studying probabilities and I have encountered this inequality. I'm trying to prove that for any random variable $X$ and any $\epsilon \gt 0$ this inequality is correct. $P(|X - EX| \ge \epsilon)...
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votes
0answers
31 views

Probability: How do I prove this inequality? [duplicate]

While studying probabilities and I have encountered this inequality. I'm trying to prove that for any random variable $X$ and any $\epsilon \gt 0$ this inequality is correct. $$P(|X - EX| \ge \...
2
votes
1answer
25 views

Confusion in Calculating Conditional Probability mass function

Question: If $X_1$ and $X_2$ are independent binomial random variables with respective parameters $(n_1,p)$ and $(n_2,p)$, calculate the conditional probability mass function of $...
-4
votes
2answers
50 views

The Executioner Conundrum [closed]

You are a military executioner tasked with eliminating some of the most dangerous criminals on Earth. You are handed 100 such criminals for immediate termination. However, just as you are about to ...
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votes
1answer
47 views

What are the odds that truck a arrives before truck b? [closed]

given truck a arrives at a random time between 9am and 11am, and truck b arrives at a random time between 10am and 12pm (noon). what are the odds that truck a arrives before truck b
0
votes
4answers
78 views

Expected numbers of boys in a family

Given $2000$ families with $4$ children in each family and suppose each sex is equally likely, what is the expected numbers of families having at least one boy? Hint: One friend tell me that $X$ ...
0
votes
1answer
28 views

Understanding the solution of finding the number of red balls drawn before the first black ball is chosen

Question: An urn contains $n + m$ balls, of which n are red and m are black. They are withdrawn from the urn, one at a time and without replacement. Let $X$ be the number of red balls removed ...
0
votes
2answers
35 views

Balls thrown into baskets randomly

4 balls are thrown randomly into two baskets (Basket A, Basket B), what is the probability that 2 or more balls will be in Basket A, I'm not sure im correct but $\Omega = 5$. because there are 5 ...
0
votes
0answers
44 views

Integral evaluation - Gamma distribution

I have a sequence of independent random variables which are $\chi^2(1)$ distributed, $(X_i)_{i=1}^n$, $X_i\sim\chi^2(1)$. If I consider the sum $\frac{t}{n}\sum_{i=1}^n{X_i}$ this should be $\sim\text{...
0
votes
1answer
34 views

Sum of hypergeometric distribution

An urn contains $w$ white and $b$ black balls. $n$ extractions without replacement are made (Hypergeometric distribution). The distribution of $\mathbb{P}(X_i=s)$ with $i\geq s$ ($s$ white on $ith$ ...
1
vote
1answer
45 views

The standard deviation is more stable than the mean?

In an introduction to the subject of hypothesis testing, a book on probability and statistics for engineering students has a statement asserting that "the standard deviation is more stable than the ...
3
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2answers
50 views

Bayes Theorem problem, from Finan #9.4: $P(A\mid B ∩ C)$

The Problem: You are given $\Pr(A) = 2/5, \Pr(A ∪ B) = 3/5, \Pr(B\mid A) = 1/4, \Pr(C\mid B) = 1/3,$ and $\Pr(C\mid A ∩ B) = 1/2$. Find $\Pr(A\mid B ∩ C)$. My work: I know that $\Pr(A\mid B) \Pr(B)...