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 ...

learn more… | top users | synonyms (1)

0
votes
2answers
38 views

What is acceptable set of values $E[ \max(X-4,0)]$?

Let $X$ be a random variable such that for $d \in [0,3] \cup [6,10]$ we have $E[\max(X-d,0)] = \frac{(10-d)^2}{20}$. What can you say about $E[ \max(X-4,0)]$ (what is acceptable set of values $E[ ...
0
votes
1answer
7 views

Rolling a die/recording findings to find if S is even,odd or both

suppose you roll a fair six sided die repeatedly and the rolls are recorded. When two consecutive rolls are identical the process is ended. Let S denote the sum of all rolls made. Is S more likely to ...
1
vote
1answer
22 views

Game of Red balls two drawings are made, which rule would you choose if playing the game, rule A or rule B?

In the game of redball two drawings are made without replacements from a bowl that has four white ping pong balls and two red ping pong balls. The amount won is determined by how many ping-pong balls. ...
0
votes
1answer
25 views

Variance of stochastic process $MA(2)$

Let $\left\{ X_t \right\} $ be a stochastic process $MA(2)$ such that $X_t = Z_t + 0.8Z_{t-2}$. Where $\left\{ Z_t \right\} $ is White Noise $WN(0,1)$. Compute variance of ...
0
votes
1answer
14 views

Calculating the probability of 2 for the sum of two dices, knowing that the sum will be even.

I'm having a problem with calculating the probability of the followig scenario. I have two normal dices (6 sides a piece). I know that the sum of the two tosses is even. I want to know what the chance ...
0
votes
0answers
5 views

How do they calculate players' chances of winning in 9-handed Hold 'em or Omaha poker with hidden information?

So Omaha poker is a card game where each player is dealt 4 private cards, and then 5 community cards are dealt in the middle, and each player makes the best possible 5 card poker hand by using 2 cards ...
0
votes
1answer
13 views

Convergence of Bernoulli distributed random variables with parameter $1/2$?

In my personal study of convergence of random variables I get stuck on this: I have random variables $X_i$ that are independent and identically distributed with $P(X_i=0)=P(X_i=1)=1/2$. We define ...
0
votes
3answers
37 views

Probability problem with cards

Problem 6. A bridge hand is dealt, so each of 4 players has 13 cards from the 52 card deck. You have 8 clubs in your hand. What is the probability that at least one of the other three hands is ...
0
votes
0answers
21 views

Game of chance probability

Peter and Paula play a game of chance that consists of several rounds. Each individual round is won, with equal probabilities of $1/2$, by either Peter or Paula; the winner then receives one point. ...
0
votes
0answers
13 views

Proof that, if $X_n\rightarrow X $ weakly and $\mathcal u_x(D)=0$, then $\mathbb Ef(X_n) \rightarrow \mathbb Ef(X)$

Proof that, if $X_n\rightarrow X $ weakly and $\mathcal u_x(D)=0$, then $\mathbb Ef(X_n) \rightarrow \mathbb Ef(X)$ $D$ is a set of discontinuous points X and $f$ is bounded, measurable. We can ...
3
votes
2answers
25 views

Probability Assignment to Intervals in $\mathbb{R}^{n}$.

Given a random variable $\bf{X}$ distributed on $\mathbb{R}^{n}$, let $F_{X}(t)$ be its distribution function. Suppose we want to find $P\left(\textbf{X} \in (\textbf{a}, \textbf{b}]\right)$. I was ...
1
vote
1answer
37 views

Counting and probability gift exchange problem

There are 50 people (numbered 1 to 50) and 50 identically wrapped presents around a table at a party. Each present contains an integer dollar amount from $1 to $50, and no two presents contain the ...
0
votes
2answers
24 views

theory of probability question [on hold]

There is a lottery. From 10,000 people only 100 win, so the probability to win is 1%. Question: what is the probabily to win if you join/buy ticket to the same lottery 100 times? I am sure you can ...
0
votes
0answers
19 views

The dependence of two events and their conditional probabilities

Suppose we have sample space={A,B,C}. m(A)=1/2, m(C)=1/6, also we assume E={B,C} and F={C}. My question is are events E and notF independent events? Also I wanna know P(E|notF) and P(notF|E). ...
0
votes
0answers
5 views

Which department or agency has the lowest mean monthly earnings?

(b) Which department or agency has the lowest mean monthly earnings? I choice Department of the Army (is correct) What is the mean monthly earnings for this department or agency? (Round your ...
0
votes
1answer
10 views

Velocity Obstacles — Probabilistic Collision Cone concept

I have been working with the Velocity Obstacles concept. Recently, I came across a probabilistic extension of this and couldn't understand the inner workings. Source: Recursive Probabilistic ...
1
vote
1answer
323 views

Solving Probability Density Function for continuous random variable

The probability density of a random variable $x$ is $$f(x)=a\ \cdotp x^2\ \cdotp \mathrm{e}^{−kx}\ (k>0,\ 0\leq x\leq \infty)$$ Then, the coefficient $a$ equals $$(i)\frac{k^3}{2}\ \ \ \ (ii)\ k^3 ...
-2
votes
0answers
10 views

Inscribed and circumscribed polygons [on hold]

Given a circle, prove (with basic geometric methods: no trigonometry) that the area of any inscribed irregular polygon is strictly smaller than the area of any circumscribed polygon. Extra ...
0
votes
0answers
13 views

Drawing numbers with replacement

is the answer 1 - (1/3)^5 because Bob winning is 1/3 ^5
0
votes
1answer
49 views

Probability problem, MIT and Harvard

A student is applying to a PhD program in Computer Sciences at Harvard and MIT. He estimates that he has two out of three chances of being accepted at Harvard and two out of five chances at MIT. He ...
1
vote
1answer
17 views

choosing random numbers between 0 and 1

A and B choose random numbers x and y, b/w 0 and 1. consider : a = {abs value of the difference of the two numbers is at most 1/3} b = {none of the numbers exceeds 2/3} find P (a), p (b), p (A ...
1
vote
0answers
20 views

Probability of a coin falling on the edges of a square

Let a coin be randomly (and uniformly) dropped onto a square on the floor. Assume the edge length of the square to be $ d $ and the radius of the coin to be $ r < d/4$. I know that the probability ...
0
votes
1answer
16 views

Conditional Expectation of Binomial Given $X \leq x$

Are there any neat formulas to reduce something like $\sum_{i=0}^{x} i \binom{n}{i} p^i (1-p)^{n-i}$ where $x<n$? This would be proportional to $\mathbb{E}(X\leq x)$ where $X$~$\text{Bin}(n,p)$. ...
0
votes
1answer
56 views

Probability in DNA segmentation

I have formulated these questions ss part of a research in medical science (DNA segmentation): A series of $M$ identical balls is arranged on a line. A partition is formed by placing a stick to ...
0
votes
1answer
29 views

Probability picking socks

There are 9 pairs of socks. We choose 5 socks at random. what is the probability of getting at least 1 pair? I computed the probability of the complement and came up with $$ 1 - \frac{18\cdot ...
2
votes
1answer
33 views

Simple Question about Almost Sure Convergence

If I can show for some event $A_n$, $$ \sum_n P(A_n) < \infty. $$ Then by First Borel-Cantelli Lemma, I get $P(A_n \; i.o.) = 0$; But I still confused about how this infinitely often connects to ...
2
votes
2answers
57 views

Probability of a player winning again after i games

I'm in a computer algorithms course and have a question about basic probability. My math background includes no more than discrete math and a little calculus, so this probability question left me ...
4
votes
2answers
3k views

Prove that the sample median is an unbiased estimator

My book says that sample median of a normal distribution is an unbiased estimator of its mean, by virtue of the symmetry of normal distribution. Please advice how can this be proved.
1
vote
2answers
54 views

Variance deduction

the definition of variance is $V(X) = E((X-E(X))^2 )$ For a discrete random variable: if we have put $Y = g(X)$ , where $g$ is a real function $E(Y) = E(g(X)) = \sum\limits_{k} g(k)p_X(k)$ , ...
0
votes
2answers
29 views

Mutually exclusive events are also independent??

Mutually exclusive events are also independent or not ? can some one explain with an example? Is it compulsory for independent event to be mutually exclusive? what are the relation between both ?
0
votes
1answer
247 views

Probability - rolling a fair die 10 times, what is the probability you would match a separate set of 10 numbers?

Having some trouble with this problem... Say someone is rolling a fair die 10 times, and using that roll as an attempt to guess what number (1-6) someone else has written down on a piece of paper for ...
2
votes
2answers
27 views

Relationship “finite mean” <-> "absolutely integrable

What is the relationship between the property of a random variable (i.e. a measurable function defined on some probability space) being absolutely integrable, i.e. $$\mathbb{E}|X|<\infty$$and ...
0
votes
0answers
42 views

If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$? [duplicate]

If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$? This is not a homework problem but rather a question I had. If it is not true, what are the ...
0
votes
0answers
11 views

variations of chernoff bound

I could not get the proof for this theorem(one form of Chernoff Bound). Please help me out. Let y_1 , . . . , y_n be n independent Bernoulli random variables each with probability of success 1/2 ...
0
votes
1answer
16 views

Probability of winning after n games [duplicate]

Say you and your friend play a coin toss game where the coin is fair and there is 1/2 chance of winning for each player. What your probability of winning the next game if you have won 5 games and your ...
0
votes
1answer
18 views

Finding PMF of C by given expected value

Suppose that a cellular phone costs 20 \$ per month with 30 minutes of use included and that each additional minute of use costs 0.50 $. If the number of minutes you use the phone in a month is a ...
0
votes
1answer
20 views

Calculate ways to represent 6-bit binary without any two contiguous 1s

How many binary numbers can be represented using a 6-bit number that does not have two contiguous 1s? For example : "101010" does not have two contiguous 1s. I would love to know which ...
1
vote
1answer
21 views

Maximum load is $O(\log\log n/\log\log\log n)$

There are $n$ bins labeled $0,1,\ldots,n-1$, and $\log_2n$ players. Each player chooses a starting location $k$ uniformly at random, and places one ball in each of the bins $$k\bmod n,k+1\bmod ...
4
votes
3answers
67 views

Show that $P(X=c)=1 $for some constant c

Suppose $X$ and $Y$ are independent random variables, also $X$ and $X-Y$ are independent. Prove that $$P(X=c)=1$$ for some constant c. I tried using moment generating function, please give me some ...
0
votes
1answer
17 views

normal probability distribution

If I as just installed 1400 new lightbulbs with an expected mean lifespan of 60 months and a lifespan standard deviation of 10 months. How many bulbs will need to be replaced after 44 months? I ...
0
votes
1answer
20 views

Prove that for a sequence of people sets $S_1,…,S_d$, $\Delta_i \not = 0$ for all people

We have $k$ people $p_1,...,p_k$, and $d$ people sets $S_1,...,S_d$, where the sizes of $S_1,...,S_d$ can vary between $1$ and $k$ (so each $S_1,...,S_d$ is a set of some people from ...
1
vote
1answer
9 views

Poisson approximation to bound probability of balls in different bins

Suppose $n$ balls are thrown randomly and independently into $n$ bins. What is an upper bound that all balls land in different bins using Poisson approximation? The exact probability is $n!/n^n$, ...
1
vote
1answer
11 views

Last two bins have same number of balls

If we throw $n$ balls independently and randomly into $n$ bins, what is the probability that the last two bins have an equal number of balls? We can write that as the sum of the probability that each ...
6
votes
2answers
593 views

Regression towards the mean v/s the Gambler's fallacy

Suppose you toss a (fair) coin 9 times, and get heads on all of them. Wouldn't the probability of getting a tails increase from 50/50 due to regression towards the mean? I know that that shouldn't ...
1
vote
0answers
28 views

Application of Law of Iterated Expectations

Could you please explain something from a text I am reading? We're given that $E(\epsilon_i)=0$ and $E(\epsilon_i x_i)=0$ for $i\geq 1$. Write $g_i=\epsilon_i x_i$ and we further know that $$ ...
1
vote
3answers
28 views

Solving for n in the exponent.

Well, it's another question I feel like I should know. I'm trying to model the number of successes before the first failure. The probability of successes is given as $p$, which makes the probability ...
4
votes
1answer
278 views

Different Perspectives of Multinomial Theorem & Partitions

There are 2 important interpretations of the multinomial theorem and coefficients. 1: Determining the number of ordered strings that can be formed using a set of letters. For example, with 1 m, 4 ...
0
votes
0answers
4 views

Maximum diagonal entry of a multivariate normal sample covariance matrix

Let $\Sigma$ be a covariance matrix of $n$ data points in $\mathbb{R}^p$. So $\Sigma$ is $p\times p$. Suppose that the $n$ points are drawn from the distribution ...
26
votes
1answer
479 views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
0
votes
1answer
9 views

A property of conditional expectation

Given a probability space $(X , \mathcal{M} , m)$ and $\mathcal{A}$ is a $\sigma$ sub algebra of $\mathcal{M}$. Let $\mathbb{E}$ be the condition expectation given $\mathcal{A}$. Given $f$ is an ...