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 views

Moment generating function of sample mean of bernoulli random variables

Let $p \in (0,1)$ and $n \in \mathbb{N}$. We consider a sample of $n$ i.i.d. Bernoulli variables $X_1,\dots,X_n$ with parameter p. Computer $E[e^{\lambda\bar{X_n}}]$ such that $\bar{X_n}= \frac{1}{n} ...
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0answers
4 views

How to do the inverse problem of the kernel density estimation

Given $x_1, x_2,..x_n ; x_i \in R$ that drawn from an unknown distribution $P(x)$ and a constant $ C$ $ 0 \leq C \leq 1$. Find $x^{*}$ such that $$P(x^{*}) =C$$. We want to use the kernel density ...
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2answers
21 views

Probability of getting 5 heads on 10 (fair) coin flips?

Even before attempting the problem, I immediately defaulted to an answer: 1/2. I thought that this was a possible answer since the probability of flipping a head on one flip is definitely 1/2. I ...
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1answer
13 views

If I have that $\limsup_{n}E|X_n|^{r} \leq E|X|^{r}$, is that enough to show that $\{|X_n|^{r}:n\geq 1\}$ is uniformly integrable?

If I have that $\limsup_{n}E|X_n|^{r} \leq E|X|^{r}$, is that enough to show that $\{|X_n|^{r}:n\geq 1\}$ is uniformly integrable? I am not sure here if the limsup condition here is as strong as if I ...
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2answers
52 views

Modeling with Markov Chains and one-step analysis

I have set up the following model: Let $X_n$ be the number of heads in the $n$-th toss and $P(X_0=0)=1$. I can calculate the transition matrix $P$. Define $$ T=\min\{n\geq 0\mid X_n=5\}. $$ Then ...
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1answer
38 views

How to calculate $x\%$ chance of success?

Everything I've looked for points to Binomial Distribution, but I have no idea how to use it. Basically, I have $2$ sets of $7$ rolls. Each roll has a specific chance of success. Set 1: ...
2
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0answers
31 views

Stochastic process on compact spaces

I just heard some strange reasoning that I would like to understand with your help, let me describe the situation (unfortunately, I hesitated to ask the lecturer about it, because I apparently lacked ...
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1answer
15 views

Conditional Independent clarification

Let's say I have $3$ events with probabilities $P(A) = 0.5, P(B) = 0.5$ and $P(C)= 0.5,$ and I need to find if $$P(A \cap B \mid C) = P(A \mid C)P(B \mid C)$$ I am tying to prove this by expanding ...
2
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2answers
30 views

Probability of island having 8 people born with disease, estimate?

The chances of being born with a certain disease are estimated as $1$ in $1200$. What is a good estimate of the chance that an island with $10000$ inhabitants has precisely $8$ people born with that ...
2
votes
1answer
26 views

Are these two events $A$ and $B$ independent?

Abe and Bernard are dealt five cards each from the same $52$ card deck. Let $A$ be the event that Abe gets a flush (five cards of the same suit) and $B$ be the event that Bernard’s five cards are of ...
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1answer
19 views

What is the probability that these two objects are of the same color?

We have $11$ bins with $10$ objects each. Every object is either black or white, and the $i$th bin ($1 \le i \le 11$) has precisely $(i -1)$ black objects in it. Someone selects, uniformly at random, ...
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1answer
15 views

Three people have been exposed to a certain illness. Once exposed, a person has a 50-50 chance of actually becoming ill. [on hold]

Three people have been exposed to a certain illness. Once exposed, a person has a 50-50 chance of actually becoming ill. a) What is the probability that exactly one of the people becomes ill? I am a ...
4
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1answer
20 views

$p(X)$, $P(Y)$, $p(Z) > 0$ and every pair of these events is independent, then $p(X \wedge Y \wedge Z) > 0$?

Is the following statement true or not? Let $X$, $Y$, $Z$ be $3$ events in the same sample space such that $p(X)$, $P(Y)$, $p(Z) > 0$ and every pair of these events is independent. Then $p(X ...
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2answers
66 views

Calculating the sample standard deviation

I've been preparing for an exam tomorrow and was doing a past paper when it appears the mark scheme has an answer that I just cannot fathom. The exact question is to "find a 95% confidence interval ...
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2answers
413 views

Probability of winning blackjack dice game?

I know a little bit about probability but I am not sure how to calculate this: In a dice game of blackjack, there are two parties. The player and the dealer. The aim of this game is to get as ...
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2answers
46 views

Defining the states when we roll one single die repeatedly

We roll a single die and the game stops as soon as the sum of two successive rolls is either 5 or 7. We want to find the probability that the game stops at a sum of 5. It seems like Markov ...
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0answers
8 views

probability that a customer who purchases up to $5$ songs from $4$ music genres prefers jazz and buys at least $3$ songs [on hold]

Customers can choose from $4$ music genres: jazz, rock, new age, country; and can purchase up to $5$ songs. The events are: $A =$ customer prefers jazz and buys at least $3$ songs $B =$ the customer ...
0
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2answers
16 views

Conditional probability with dependent events

We have 2 dice. One is fair. The other one lands by the following probabilities: 6: 1/2 5: 1/10 4: 1/10 3: 1/10 2: 1/10 1: 1/10 We roll both dice. What is the ...
3
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3answers
30 views

What is the probability that a randomly chosen positive three-digit integer is a multiple of $7$?

What is the probability that a randomly chosen positive three-digit integer is a multiple of $7$? Is my answer right?: $$\frac{100}{7} = 14 , \qquad \frac{999}{7} = 142$ Then there are $142 - 14 = ...
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0answers
14 views

integral and probability

Let $N_t$ be a Poisson process and $S_{N_t}=X_1+...+X_{N_t}$. Let $A_t=t-S_{N_t}$ and $B_t=S_{N_t}-t$ 1) Show $P(B_t \geq x \ \text{and}\ A_t \geq y)=\frac{1}{E(X_1)} \int_{x+y}^{\infty} P(X_1 \geq ...
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1answer
17 views

Conditional Probability clarification

Here's a sample problem: Before each workout, I either drink a cup of coffee, a gatorade, or a cup of water. The probability of coffee is P(C) = 0.6, the probability of gator is P(G) = 0.3, the ...
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1answer
595 views

Intuition on Harris recurrence

I am trying to get some intuition on Harris recurrence in Markov chains. Define state space $\mathcal S$ comprising a single communication class, $f_{ii}^{(n)}=P(X_n=i, X_{n-1}\ne i,\ldots X_1\ne ...
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1answer
17 views

Win/Lose ratios and selection strategies

Imagine the following scenario: You're on a TCG tournament which allowed you to bring N decks with you. After each game, you might select another deck for your next game. You are allowed to keep ...
1
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2answers
16 views

Exponential distributions [on hold]

Good evening to all, I'm so much confused about a question; Assume there is a workshop with two machines. The times until the failures of machines $1$ and $2$ are independent and exponentially ...
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1answer
100 views
+50

How do you find the probability of A winning if the probability of getting a favourable outcome in the $r^{th}$ turn is a function of $r$?

Problem: Two players A and B are playing snake and ladder. A is at 99 and he needs 1 to win in rolling of a dice. However, he is always allowed to re-throw the dice if 6 appears. What is the ...
2
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2answers
22 views

chain rule conditional entropy

I have to prove the chain-rule for conditional entropy. I kept getting stuck on one step, so I looked up a proof and found this: \begin{align}H(Y\mid X)&= \sum_{x\in\mathcal X, y\in\mathcal ...
0
votes
1answer
18 views

Find the probability of B

Suppose you roll a fair 6-sided dice three times. There are $6^3$ possible outcomes and each is equally likely. Let $A_1, A_2, A_3, A_4, A_5,$ and $A_6$ be the events that the last value is ...
0
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0answers
29 views

An urn contains $nr$ balls numbered $1,2..,n$

An urn contains $nr$ balls numbered $1,2..,n$ in such a way that $r$ balls bear the same number $i$ for each $i=1,2,...n$. N balls are drawn at random without replacement. Find the probability that ...
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1answer
24 views

What is the probability of unions of intersections?

Suppose we have two unions of (possibly overlapping) events. Let me denote the unions as: $$A = IE_A^1 \cup \dots \cup IE_A^{k_A}$$ $$B = IE_B^1 \cup \dots \cup IE_B^{k_B}$$ Each $IE_X^y$ is a ...
0
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1answer
33 views

Should I use law of large numbers or Chebyshev inequality?

I think the answer is zero. Can anyone tell me whether I should use Weak Law of Large Numbers or Chebyshev inequality . I just need a hint how to proceed. Is my answer 0 correct? Thanks link to ...
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1answer
40 views

Show that we always have $Y + Z = X + 4$.

Let $X$ be a Geometric random variable with parameter $p =\frac{1}{2}$. We define another random variable $Y$ in terms of $X$ as follows. $Y = \min\{X,4\}$ Here $\min\{X,4\}$ is the minimum between ...
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1answer
30 views

A coin is tossed $m+n$ times. Find the probability of getting atleast $m$ consecutive heads

A coin is tossed $m+n$ times. Find the probability of getting atleast $m$ consecutive heads I already know that the exact same question has already been answered here But I am trying to solve it ...
0
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1answer
24 views

Equality in Conditional Jensen's Inequality

Conditonal Jensen's Inequality says that for a convex function $\varphi$, a random variable $X$, and a sub-sigma-field $\mathcal{F}$, $E[\varphi(X)\mid \mathcal{F}] \geq \varphi(E[X\mid ...
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2answers
12 views

Possible orderings when the items are not unique?

First of all, I'm sure this question has been answered somewhere on the web, but I am just starting probability and I don't have the vocabulary to know what to look for, which is why I am asking here. ...
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0answers
7 views

Is the result of a Monte-Carlo simulation of a continuous function and with continuous input distributions again continuous?

Is the result of a Monte-Carlo simulation of a continuos function and with continuos input distributions again continuous? Suppose, we have a continuos function $f$ and a number of continuous random ...
1
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1answer
29 views

Probability to fit $1950$ items in a box that hold $1880$. Risk $5$%

If I buy $1950$ plates to fill a box that hold $1880$ what is the probability that $1950$ is enough if the risk of dropping a plate is $5$% per plate? The answer is $F_z(-2,81)=0,0025$ I just don't ...
1
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2answers
44 views

What is the probability that at least $2$ out of $4$ digits in a code is the same?

If I choose $4$ digits for a code randomly out of the digits $0$ to $9$. What is the probability that at least $2$ of these digits are the same? By at least I mean that you have to count with the ...
1
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1answer
19 views

Conditional Probablity for two independent events(Formula)

Let there be two independent events $A$ and $B$. To calculate the probability (for a particular condition) we have two relations. $P(A \cup B)=P(A)+P(B)-P(A \cap B)$. $P(A/B)P(B)=P(A \cap B)$, i.e., ...
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3answers
24 views

{students 1 and 2 are in different groups} vs {students 1, 2, 3, and 4 are in different groups}

Source: Example 1.11, p 26, *Introduction to Probability (1 Ed, 2002) by Bertsekas, Tsitsiklis. Hereafter abbreviate graduate students to GS and undergraduate students to UG. Example 1.11. A ...
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0answers
18 views

On the probability distribution of iterated permutations

I have this little problem that has been nagging me for a couple of months now. It occurred to me when considering the fairness of card shuffling methods. Here's my best attempt at formalizing it: ...
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0answers
28 views

Mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$?

Is the mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$? In particular, in the extreme case that the pairwise mutual informations are ...
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2answers
21 views

What is the probability or percentage or frequency by which hello line will be printed?

I have a below method which is called every one minute from background thread and that background thread keeps running always. ...
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1answer
67 views

Equivalence in conditional probability

I am wondering the equivalence of the following problem. When we computing $$ P(\mathbf{x}_1 | \mathbf{x}_2, \mathbf{x}_3) $$ is it equivalent as following, at first define $\mathbf{y} = ...
0
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3answers
28 views

Is every bounded sequence of random variables in $L^1$ convergent? [on hold]

If $\{X_n\}_{n>0}$ is a bounded sequence of random variables is it true that $E(X_n)$ converges?
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0answers
21 views

Destined pair 'guessing' game

n people participate in a game. Before the game the participants are put into random secret 'destined' pairs. Each round the participants pick1 their own pairs and ...
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0answers
23 views

probability,calculus

Let $N_t$ be a Poisson process and $S_{N_t}=X_1+...+X_{N_t}$. Let $A_t=t-S_{N_t}$ and $B_t=S_{N_t}-t$ 1) Show $P(B_t \geq x \ \text{and}\ A_t \geq y)=\frac{1}{E(X_1)} \int_{x+y}^{\infty} P(X_1 \geq ...
0
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1answer
29 views

Need Help with continuous random variable probability problem [on hold]

Suppose that an electric device has a life length $X$ which is considered as random variable with pdf: $f(x)=e^x$, $x>0$. Suppose that the cost of manufacturing one such item is $2$. The ...
2
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0answers
41 views

Expectation and Variance of $X/(X+Y+Z)$

I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal ...
0
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1answer
26 views

Continuous probability - calculate probability of r.v and distribution function

This is the question: $X$ is a continuous random variable whose probability density function is given by $$f(x)=\begin{cases} \frac{1}{9}x^2 & \text{if $0\leq x \leq 3$}.\\ 0 ...
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1answer
39 views

How to show a sequence of independent random variables do not almost surely converge by definition?

I have a sequence of independent random variables $X_1, X_2, \ldots$ where $$ X_n = \begin{cases} 1 & \quad \text{with probability} \ 1/n \\ 0 & \quad \text{with ...