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
votes
1answer
40 views

Is this proof of convergence in probability correct?

${X_i}, i = 1,2,\dots$ i.i.d random variables, and $S_n = \sum_{i=1}^nX_i$ is defined as partial sum as usual. If $\frac{S_n}{n} \to 0 \quad $ in probability show that $$\lim_{n\to \infty} \min_{...
0
votes
2answers
46 views

Probability of space junk hitting the southern hemisphere [closed]

About $20\%$ of the southern hemisphere is land. South Africa takes up about $5\%$ of the the land surface of the southern hemisphere. A piece of space junk is falling to earth such that it could hit ...
1
vote
0answers
35 views

Bounded Poisson Random Walk

Suppose we have the following random walk. Let $X_0=\lambda \in \mathbb{N}$ and then $ \begin{cases} X_t=\operatorname{Pois}(1) & \text{if } X_{t-1}=0 \\ X_t=\operatorname{Pois}(...
1
vote
1answer
66 views

Conditional Probability Exercise - Car Bomb

I've been trying to work out this exercise on conditional probability, and it has me completely stumped. The question is as follows: Consider a garage containing $n$ cars. Exactly one of the cars ...
3
votes
0answers
47 views

How to approximate the cumulative distribution function of the normal by a product of functions?

Suppose, there are $n$ vectors $\mathbf{X}_1$, $\mathbf{X}_2 \ldots \mathbf{X}_n$ of unequal lengths which can be combined to a new vector as $$ \mathbf{X} = \begin{bmatrix} \mathbf{X}_1 & \mathbf{...
1
vote
1answer
32 views

Combined probability of hit in look up tables with some common index bits

Consider two tables A and B consisting of $l_a$ and $l_b$ counters respectively - $l_a$ and $l_b$ are powers of two and the counters are initialized to zero. Each table has its own index ...
2
votes
0answers
31 views

Sum of two logarithmic random variables

I would like to compute the PDF of the difference of the logarithms of two shifted Rayleigh laws ($Z$): \begin{equation} Z = \log{X_{1}} - \log{X_{2}} \end{equation} where $X_1 \sim R(\alpha_1, \...
0
votes
2answers
81 views

A fair die is rolled 100 times. Which of the following has a probability of at least 95%?

A fair die is rolled 100 times. Which of the following has a probability of at least 95%? $ $ 1.) Sum of the rolls is greater than 322 2.) Sum of the rolls is less than 392 3.) Number of rolls ...
2
votes
1answer
20 views

Probability: Find Dispersion of X + Y

$X = \operatorname{Bi}(3,\frac14), Y=\operatorname{Bi}(4,\frac12), \operatorname{Cov}(X,Y) = -\frac34.$ Dispersion of $X+Y =?$ $D(X) = npq = \frac9{16}. D(Y) = npq = 1$ $D(X + Y) = D(X) + D(Y) = \...
1
vote
0answers
48 views

walking randomly in $1D$, $2D$ and $3D$ place [duplicate]

In our math book it is written that if you want to go to a random point in a $1D$ place and you randomly go right and left and also you have unlimited time finally you arrive at that point. Also in a $...
1
vote
1answer
27 views

Probability of 3 different factors

As a probability newbie, I'm struggling to understand a basic principle in a question I'm facing, which is: Researchers are trying to study the following factors for one's success in school: ...
4
votes
0answers
78 views

Conditional expected value of mutlitple draws from uniform distribution

There are $m$ i.i.d. draws of $x$ made from a uniform distribution on $[0,1]$. The $n$ ($n\leq m$) lowest draws are "winners", i.e. if we write $x_1\leq\ldots\leq x_n\ldots\leq x_m$, the draws $x_1$ ...
4
votes
1answer
132 views

Is the Fermat primality test secure enough for very big numbers?

The random variable $X_m$ is the number of trials before $n\notin\mathbb P\wedge n|2^{n-1}-1$ where $n$ is an odd random integer $2^{m-1} < n < 2^m$. Computer simulations makes me believe ...
0
votes
1answer
81 views

weak L1 convergence

Given a sequence $Y_{un}$, where $Y_{1n},Y_{2n},\ldots$ have the same domain. Assume for every $u\in \mathbb{N}$ we have $e^{itY_{un}}\rightarrow \mathbb{E}[e^{it M}]$ weakly in $L_1$ as $n\rightarrow ...
0
votes
0answers
26 views

Multi armed bandit problem how to calculate $\mathbb{P}(A > B)$ using Thompson sampling [closed]

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 : ...
2
votes
1answer
49 views

Probability: Even sum

Out of a set of $\{1 ... 2n\}$ integers, we choose a sequence of length $n$, of integers from the set. (An element can be selected twice, order matters). What is the probability that the sum of the ...
0
votes
1answer
22 views

Moment generating function (MGF) of the ratio distribution $\displaystyle\frac{X}{Y}$

If we know the moment generating functions (MGFs) of the random variables $X$ and $Y$ to be $M_{X}(s)$ and $M_{Y}(s)$, respectively. The MGF of the sum $X+Y$ will $M_{X}(s) \cdot M_{Y}(s)$. So what ...
0
votes
1answer
43 views

Probability book choosing questions

So I am doing homework and have the following question If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems, and a dictionary. What is the probability that (a) the ...
0
votes
2answers
38 views

Expected value of colors picked from basket

I have a basket with 4 balls with different colors. What is the expected value of distinct colors I can see after picking 4 times from bin. I return the ball back after each try. I tried computing ...
2
votes
0answers
26 views

Chebyshev's inequality and quadratic function

I am trying to use Chebyshev's inequality in order to find sample sizes $n$ such that some condition is met with probability $p$ or larger. That is, find $n\in \mathbb{N}$ such that $\mathbb{P}(X_n &...
1
vote
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
88 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
vote
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
vote
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
52 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
57 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
102 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
vote
1answer
49 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
43 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
vote
0answers
35 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
38 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
vote
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
149 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
34 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
61 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 ...