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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0answers
7 views

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

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
votes
0answers
8 views

Conditional Independent clarification

Let's say I have 3 probabilities. A = 0.5 B = 0.5 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 the formula above to: ...
0
votes
0answers
13 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 ...
0
votes
2answers
15 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 ...
1
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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 ...
0
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0answers
24 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 ...
2
votes
1answer
16 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
14 views

Exponential distributions

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 ...
0
votes
1answer
17 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
votes
1answer
25 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 ...
2
votes
2answers
20 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 ...
1
vote
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. ...
0
votes
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 ...
0
votes
1answer
20 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 ...
0
votes
0answers
15 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: ...
3
votes
3answers
28 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 = ...
1
vote
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., ...
0
votes
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. ...
0
votes
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 ...
0
votes
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
votes
3answers
27 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?
0
votes
1answer
31 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 ...
0
votes
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
votes
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
votes
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 ...
1
vote
1answer
38 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 ...
1
vote
1answer
27 views

Door Prizes - Probability [on hold]

Joe goes to a party with three friends. There is a drawing for four door prizes. Each person has an equal chance of wining a prize. No one can win more than one prize. If there are totally thirty ...
0
votes
1answer
20 views

How to show convergence in probability by just using the definition?

I have a series of random variables $X_1, X_2, \ldots$ where $$ f(X_n) = \begin{cases} 1/n & \quad \text{if} \ X_n = 1 \\ 1-1/n & \quad \text{if} \ X_n = 0 \\ 0 & ...
2
votes
2answers
45 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 ...
2
votes
1answer
36 views

What is the intuitive difference between almost sure convergence and convergence in probability?

It is a standard fact in probability that almost sure convergence is stronger than convergence in probability. I can only see the differences in the proof. However, is there a way to view it ...
0
votes
1answer
19 views

How can I compute the mean of a sequence of random variables?

Suppose that I have a sequence of random variables where $X_1, X_2, \ldots$ where the pdf of $X_n$ is equal to: $$ f_n(x) = \begin{cases} (n-1)/2 & \quad \ -1/n < x < 1/n \\ ...
0
votes
1answer
30 views

Continuous Probability - Bus Arriving

I am trying to do the following question: Number 24 and number 42 buses arrive independently at the corner of Mayeld Road at a random rate of 3 and 4 per hour respectively. You arrive at the ...
3
votes
1answer
16 views

Moment generating function and convergent random variables

denote by $X$ and $X_n$, $n\in \mathbb{N}$, random variables and $r\in\mathbb{R}_+$ with $E=\mathbb{E}\left[ e^{rX} \right] < \infty$ and $E_n=\mathbb{E}\left[ e^{rX_n} \right] < \infty$ for all ...
0
votes
0answers
24 views

Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...
0
votes
0answers
25 views

Probability to get from A to C.

There has been a snowstorm and Bob is trying to drive from A to C. p and q are the probabilities that the two roads are passable. What is the probability that Bob can get from A to C? Note that ...
1
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0answers
37 views

Single, 6-sided die probability

I'm working on an assignment and I'm more or less new to stats. It might be the wording of the questions that's getting me as well. It deals with a regular 6-sided die. 1.a) What is the mean number ...
0
votes
0answers
22 views

poisson process(exercice)

Let $N_t$ 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 $P(B_t \geq x \ and\ A_t \geq y)=\frac{1}{E(X_1)} \int_{x+y}^{\infty} P(X_1 \geq u)du$ with ...
0
votes
1answer
14 views

Finding the MLE for an open interval.

So the problem says: Let $X = (X_{1},...,X_{n})$ be a random sample, where $X_{i} \sim Unif (0, \theta _{0})$, where $\theta _{0} \in (0,\infty)$ is unknown. Find the maximum likelihood estimator $T$ ...
0
votes
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 ...
0
votes
0answers
11 views

Conditional probability,two conditions

A doctor operates on patient with a certain disease if he is 80% sure that he has it.We have a patient for whom the doctor is 60% sure that he has the disease,so he makes him do another test which is ...
0
votes
1answer
10 views

Test predictability with Bayes' Theorem

Say we have a disease and a test for it. P(A :=a person has the disease)= 0.01 ( example) P( B:=test is positive | A )=0.95 Is this enough information to calculate the probability that a person has ...
3
votes
1answer
46 views

How to find $z$-score

I have some probabilities, but I have to find the $z$-score. I am not sure how do to this when I am told I have to use slope-intercept. Where do I plug the numbers in exactly? Here is one of my ...
0
votes
0answers
12 views

Stationary process vs stationary increments

Am I right that these are not the same, i.e. a stationary process need not have stationary increments and vice versa? example: Brownian motion is not a stationary process but it has stationary ...
0
votes
4answers
42 views

Probability of second card being an ace

I have this task about cards: Consider choosing a card from a well-shuffled standard deck of 52 playing cards. Suppose that, after the first extraction, the card is not reinserted in ...
1
vote
0answers
11 views

renewal process and Markov property

Let $A_t=t-S_{N_t -1}$ with $N_t$ a renewal process 1)Show $A_t$ checks the Markov property my proof: $S_{N_t}=X_1+\cdots+X_{N_t}$ and the increments are independents $$P(S_{N_t-1}=t-y\mid ...
1
vote
0answers
15 views

Range of a standard brownian motion, using reflection principle

With a standard brownian motion $B_t$, I'm trying to find the distribution of the "range": $$R_{t} = \sup_{0 \leq s \leq t} B_s - \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$ The ...
2
votes
1answer
12 views

Inverse of Gaussian CDF, Sum

Consider the following setting. Let $k = 1, \ldots, n$ and define $$y_k= \Phi^{-1}\left(\frac{k}{n+1}\right),$$ where $\Phi$ is the inverse of the CDF of a standard normal. I noticed numerically ...
1
vote
1answer
46 views

Show $ (\int_{-\infty}^\infty \sqrt{p}\sqrt{q}d\mu)^2\leq 2 \int_{-\infty}^\infty \min\{p,q\}d\mu $

Consider a random variable $X$ in $(\Omega, \mathcal{F}, \mathbb{P})$. Let $p,q$ be two densities with respect to a measure $\mu$ in $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where ...
0
votes
1answer
49 views

The normal distribution - how to calculate the integral

Question: It was found that the mean length of $100$ parts produced by a lathe was $20.05$ mm with a standard deviation of $0.02$ mm. Find the probability that a part selected at random would ...
3
votes
1answer
20 views

Probability of working machine with $3$ components

I have this task to do: A machine is composed of $3$ components, which function independently of each other with probabilities $p_1$, $p_2$ and $p_3$, respectively. Calculate the probability ...