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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3
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1answer
139 views

If $X_n = Y_n + Z_n$ in distribution, and $X_n$ and $Y_n$ converge in distribution, does $Z_n$?

The random variables take values in $\mathbb{R}^d$. I have tried to prove this using characteristic functions. Let $\hat{\mu}_{X_n},\hat{\mu}_{Y_n},\hat{\mu}_{Z_n}$ be the characteristic functions of ...
0
votes
1answer
54 views

Probability that two numbers in different numeral system match a pattern

How likely is it that the following holds true. $ s_1=\sum_{i=0}^{9}{16^{i} \cdot x_i}$ where $ x_i\in[0,15]\cap\mathbb{N}$ and $x_i=0$ $\forall$ $i \in\{7,6,3,2\}$ $ s_2=\sum_{i=0}^{7}{36^{i} ...
0
votes
3answers
49 views

Expected value of random variable $X$ ~ $N(170, 25)$

Here's a question: Person's height in CMs is a random variable $X$ ~ $N(170, 25)$. Door's height is $180$ cm. What is the expected value of number of people that can enter the door until the first ...
1
vote
1answer
45 views

Let $X_n$ be the $n$-th partial sum of i.i.d. centralized rv and $\mathcal{F}_m:=\sigma(X_n,n\le m)$, then $\text{E}[X_n\mid\mathcal{F}_m]=X_m$

Let $(\Omega,\mathcal{F},\text{P})$ be a probability space $\left(Y_i\right)_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variables ...
3
votes
3answers
95 views

Choosing new teammates

My sister gave me a combinatorical riddle. It doesn't appear to be hard, but I ask you if my thoughts are right, just for certainty. Here it is. Assume you belong to a group of $100$ people, and ...
1
vote
1answer
83 views

Quiz: people and hats

I've created this quiz, but I'm not sure if the answer that I've found is correct or not. Three people meet at a pub, each of them has a blue or a red hat on his\her head. Nobody knows the colour of ...
0
votes
1answer
65 views

Expected value of $\max(x,y) $ given that $x<y$

What is the expected value of $\max(X,Y)$ given that $X<Y$? $X$ and $Y$ are independent and exponential random variables.
2
votes
1answer
41 views

Count the number of one pair hands in a standard deck

In an attempt to answer this question, I tried the following solution: Let $A$ denote the number of one-pair hands Let $B$ denote the number of two-pair hands Let $C$ denote the number of ...
0
votes
2answers
102 views

Null Hypothesis at 5% level of significance

When a null hypothesis is accepted at 5% level of significance, what is the probability that the decision made is wrong?
0
votes
0answers
37 views

Possible typo in the proof of a proposition?

My books states the following proposition: Suppose $( A_{\alpha} )_{\alpha \in \Gamma} $ is a collection of pairwise disjoint events and $P$ a probability such that $P(A_{\alpha}) > 0 $ for all ...
0
votes
2answers
42 views

How can I show the following sequence converge to $0$?

Suppose $\{ A_n \}_{n \geq 1} $ is a sequence of pairwise disjoint events and $P$ a probability. Im curious as to why $$ \lim_{n \to \infty} \sum_{k=n}^{\infty} P(A_k) = 0 $$ I know that $$ ...
3
votes
2answers
47 views

Probability of a 2 pair in poker

What is wrong with my method of find the probability of a 2 pair? For the first card we have 52 choices, one this is picked( call it A) we have 3 choices(different suite than A but this same value) ...
1
vote
1answer
53 views

How can I can calculate the CDF of the random variable?

Assuming that $H$, $N$ are random variables in which $H$ is distributed following exponential distribution with mean value $\Omega$, and $N$ is a Gaussian random variable with probability density ...
1
vote
2answers
61 views

Probability married couple problem

Five married couples are randomly paired at a rectangular table. Five wives on one side and five husbands on the other side. What is the probability that only one man soy across from his wife. ...
2
votes
1answer
57 views

Central Limit theorem solving for n with absolute value

First off, I'd like to apologize for the lack of latex in this question. I'm not understanding a crucial step in a homework problem I have. It states: Suppose that a measurement has mean $\mu$ and ...
2
votes
2answers
38 views

Confidence Intervals that Contain the Mean: Designing an Activity

This past Wednesday, I had my stat class do the following exercise: Roll a fair 6 sided dice 25 times. Take the sample mean of the face value. Using the standard deviation of the uniform ...
0
votes
1answer
36 views

Odds of Winning Office NCAA Pool

I have 6 coworkers competition in a NCAA bracket. I'm trying to find out how to calculate who has the best chance of winning. For example currently the score card looks like: Player 1. Current Right ...
1
vote
1answer
31 views

how to calculate discrete event probability?

A bookstore sells children's books that belong to two publishing companies A and B and were published between years 2000 and 2004. The probabilities of a book being published by companies A and B are ...
0
votes
0answers
44 views

Probability and Wilcoxon rank sum test

The Wilcoxon rank sum test is a widely used procedure for assessing whether two groups of observations come from the same distribution. Let group 1 consist of i.i.d. X1,…, Xm with CDF F and group 2 ...
0
votes
1answer
39 views

Determining 2 events in a sample space

Consider 2 events $A$ and $B$ in a sample space $S$. Assume that $\Pr(A) = {1\over 2}$ and $\Pr(B\mid\overline A)= {3\over 5}$. Determine $\Pr(A\cup B)$ Assume that $\Pr(A \cup B) = {5\over 6}$ and ...
1
vote
1answer
53 views

$3\sigma^2 \le \mu(1-\mu)$ for unimodal probability distributions on $[0,1]$?

Let $f(x)$ be a normalized continuous probability distribution with support on $[0,1]$. It isn't hard to prove that the following inequality holds in general: $$\sigma^2 \le \mu(1-\mu)$$ where ...
0
votes
0answers
97 views

Probability question: order matters vs order doesn't matter

Consider the following problem: Two numbers are picked at random from $1,2,...,n$, what is the probability that they are consecutive if they are picked with replacement (and without replacement)? ...
0
votes
1answer
35 views

Covariance definition in terms of expected value

Covariance is defined as: $$COV(x,y)=\mathbb{E}[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])]$$ Is the following equation derived from the one above? If yes, how? By treating the expected value as an ...
0
votes
1answer
26 views

AMC12B Problem, probability

An unfair coin lands on heads with a probability of $\tfrac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the ...
1
vote
1answer
99 views

3 urns, each with 4 balls. select one ball from each

Three urns are labeled $1,2,3$. Each urn contains $4$ balls labeled $1,2,3,4$. A ball is drawn from each urn such that any ball is equally likely to be drawn. The number on the ball is compared to the ...
1
vote
2answers
62 views

4 Sided dice - flawed logic?

If we had a 4 sided dice (numbers 1,2,3,4 on the faces) and rolled it 6 times and recorded the results. What would be the probability that we rolled the same number 3 or more times? (Such as rolling ...
3
votes
1answer
74 views

Probability with flipping a coin 6 times

I have a fair coin and I flip it 6 times, here are the events: $Pr(A)$ = "the coin comes up heads at least 4 times" $Pr(B)$ = "the number of heads is equal to the number of tails" $Pr(C)$ = "there ...
0
votes
2answers
68 views

Optimizing Choice of Life Partner

In this link, Hannah Fry mentions that a mathematical argument has been made towards the probabilistically optimal strategy for picking someone to settle down with. The claim is as follows: If we ...
0
votes
1answer
36 views

Truncated normal mixture distribution and simulation

Let $X_1, X_2$ and $X_3$ are independent random variables and $X_i\sim N(\mu_i, \sigma_i^2)$. Let $Y$ be a normal mixture random variable that its cummulative distribution function (CDF) is given by ...
1
vote
1answer
37 views

Probability formula for this problem

Kindly guide me in solving the following problem. I am learning probability so, i would appreciate a bit of explanation. There are two men M1 and M2. Each man has got two buckets called A and B. ...
2
votes
0answers
43 views

Large Random Graph is Surely Connected

I'm trying to prove that for a random graph on $n$ vertices with edge-probability $p \in (0, 1)$ is almost surely connected as $n$ grows large. I've tried making an argument using the probability of ...
0
votes
0answers
41 views

Pick a random points $X$ in $(0, 0.5)$ , $Y$ in $(0.5, 1)$. What is the probability that the distance between $X$ and $Y$ is greater than $0.25$?

I got the following question: Pick a random point $X$ in $(0, \frac 1 2 )$ and pick another random point $Y$ in $(\frac 1 2, 1)$. What is the probability that the distance between $X$ and ...
0
votes
1answer
17 views

How often I have to do action X, with a $10\%$ chance of a reward, to have a $99\%$ chance of getting said reward?

In a game I play, there is a $10\%$ chance of getting the item I want when I complete a certain action. I'd like to know how often I need to do said action to be $99\%$ sure I will have this item.
0
votes
0answers
42 views

What is the probability of a repeat occurring at least once in the decimal expansion of an irrational number?

Let's take pi as an example. Pi = 3.14159... What is the probability that in the decimal expansion of pi, that there exists some number n such that that after the nth digit of the decimal expansion ...
3
votes
3answers
45 views

Showing the limit of an event is zero

Let $\{ A_n \}_{n \geq 1 } $ be a sequence of events which are pairwise disjoint and let $P$ be a probability measure. Show carefully that $\lim_{n \to \infty} P(A_n) = 0$ Attempt to the solution: I ...
0
votes
0answers
22 views

probability with binomial function questions

A die is rolled 4 times and the number of times that a six occurs is noted (a) calculate the probability that 1 six is rolled $$P(\text{exactly one 6 appears}) = {4\choose 1} (1/6)^1 (5/6)^3 = .386$$ ...
2
votes
0answers
70 views

How can I show/solve this equation?

I need help to prove the following equation. $X_n$ is an iid random variable, with: $$\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=\frac{1}{2}$$ Show: ...
0
votes
2answers
58 views

putting 4 balls into 4 boxes, each ball have a fair probability to fall into each box. Calculate the expected value of empty boxes.

Given the following question: putting 4 balls $\{ball_i\}_{i=1}^4$ into 4 boxes $\{box_k\}_{k=1}^4$. Each ball $ball_i$ have a fair probability to fall into each $box_k$, independently to ...
0
votes
1answer
36 views

In a box there are balls numbered m to n (n>m) and with returns 2013 are drawn and the numbers on them are registered..

What is the probability that the largest and smallest registered numbers are $k\ and\ l (m\leq l <k\leq n)?$ Do the assignment in the case that the balls are drawn without returns, $2012\leq k-l$. ...
0
votes
1answer
39 views

Proving that the Markov chain is recurrent - Confusion/Help

Giving the following transition matrix [ 0.9 0.1 ] [ 0.8 .2 ] Classify the states From drawing the graph I know that both stats are recurrent. However I'm really failing to prove mathematically ...
1
vote
0answers
20 views

Moment determinacy of joint distribution implies moment determinacy of projection

Let $X$ be an $n$-dimensional random vector with a distribution $P$ that is uniquely characterized by its moments. Consider the scalar random variable $\xi_1 X_1 + \dots \xi_n X_n$ with a vector $\xi ...
2
votes
1answer
55 views

How to prove this Brownian motion convergence?

Let $W_t$ be a Brownian motion. How do I show the following? $$ \alpha > \frac{1}{2} \Rightarrow \lim_{t\rightarrow\infty} \frac{W_t}{t^{\alpha}} = 0 \text{ a.s.} $$ Showing convergence of this ...
0
votes
1answer
22 views

How many events are associated with a Sample Space of $n$ outcomes?

For example if $n = 2$, the amount of events that may occur is $3$: $S = \{a, b\}$ $E^* = \{a, b, ab\}$ So what are the maximum amount of events that may be derived if the number of samples $= n$?
0
votes
4answers
47 views

Probability that exactly k men are chosen by 5 women

Five young women and three young men are friends. One night, each of the women calls one of the men, whom she chooses at random. Find the probability that exactly k men are called for k = 1,2, and 3. ...
0
votes
0answers
23 views

Central limit theorem with multiple means and variances

How does the Central limit theorem work when there are multiple random variables, each one with a different mean and variance? For example, let $X_1, X_2$ and $X_3$ be random variables that represent ...
-3
votes
1answer
39 views

The expected value of $1/(1+X)$ where $X$ has geometric distribution

Let $X$ be a random variable with a geometric distribution with parameter $p$. Calculate $\Bbb E[1/(1+X)]$ I've looked at answers online but still do not understand how to solve this. ...
1
vote
0answers
27 views

How to find covariance of sample mean and sample standard deviation

I have a question to find the covariance of sample mean and sample standard deviation based on the following: I have tried something on my scratch paper, but for some reason, I cannot upload on ...
1
vote
2answers
25 views

Probability: Determine independent event from isosceles triangle problem

A point is selected at random from the triangle {(x,y):0 <=x <= y <=1}. Let E be the event that a selected point has the x coordinate is less than 0.5 and F be the event that y-coordinate is ...
1
vote
0answers
21 views

Eigenmatrix of a Wishart Matrix

A lot of papers mention statements of the form "The Eigenmatrix of a Wishart matrix are uniformly distributed on a sphere of $P$ dimensions" but I am having a hard time finding the conditions under ...
1
vote
1answer
25 views

Derivated distribution question

I'm having trouble with the following problem: "The salary of a group of employees is represented by the continuous random variable $X$ with probability density function $f_X(x)= \begin{cases}x/2 ...