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
0answers
5 views

Question regarding double integrals

Regarding the Buffon's needle case for long needles of length $ l>t, $ (the distance between the parallel lines on the floor), we need to solve the integral $$ \int_{\theta=0}^{\frac{\pi}{2}} ...
1
vote
2answers
25 views

8 questions on true/false quiz. Expected number of correct answers with a given probability of correct answer of a question “i”.

The odds/probability of a correct answer of question $i$ is $p=1-2^{-i}$ What is the expected number of correct answers? My attempt: $$(1-2^{-1})\cdot (1-2^{-2})\cdot (1-2^{-3})\cdot ...
0
votes
1answer
28 views

Probabilities of no pair and $3$ of a kind in a $7$ card draw from a fair deck of cards. [on hold]

Recall that a standard deck of $52$ cards consists of $4$ cards in each of $13$ denominations. Suppose you are dealt $7$ cards at random. (a) What is the probability that you will have no pair ...
2
votes
2answers
30 views

How to show that the integral of bivariate normal density function is 1?

How to show the following? $$\large \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{2 \pi \sqrt{1-\rho^2}} e^{-\frac{x^2+y^2-2 \rho x y}{2(1-\rho^2)}} dx\ dy=1$$
0
votes
2answers
24 views

For a non-negative absolutely continuous random variable $X$, with distribution $F$. Why is $\lim_{t\rightarrow \infty}t(1-F(t))=0$?

So I am given a non-negative absolutely continuous random variable $X$ with distribution $F$, and density $p_X$. I am given the definition of expectation using simple functions and the survival ...
0
votes
3answers
30 views

Suppose you roll a pair of fair dice. What is the probability that the number of dots on the two dice sum to either $5$ or $10$?

Suppose you roll a pair of fair dice. What is the probability that the number of dots on the two dice sum to either $5$ or $10$? (a) $5/36$ (b) $7/36$ (c) $11/36$ (d) $4/36$ So here are the possible ...
0
votes
1answer
13 views

Probability of Normal Distribution

Let's say that 10 sumo wrestlers were to squeeze into an elevator that could only hold a max capacity of 2300 pounds. Let's say that the weight of the sumo wrestlers is normally distributed with a ...
0
votes
0answers
8 views

How to calculate $\mathbb{P}[Y\in F|X]_{\omega}$

Here I have an exercise of book: Probability and Measure of PATRICK BILLINGSLEY of conditional probability in the page 442, exercice 33.4 (b): Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability ...
2
votes
2answers
35 views

Prove $\mathsf E(N)=\sum_{i=1}^\infty \mathsf P(N\geqslant i)$

We want to prove that $$\mathsf E(N)=\sum_{i=1}^\infty \mathsf P(N\geqslant i)$$ We are given a hint that $$\sum_{i=1}^\infty\mathsf P(N\geqslant i)= \sum_{i=1}^\infty\sum_{k=i}^\infty \mathsf ...
2
votes
1answer
23 views

How to prove this convergence of a sum

How dos one prove the convergence of this sum $$e^{-m}\sum_{i=0}^\infty \frac{(mt)^i}{i!}=e^{m(t-1)}$$. I'm looking at the solution for a problem about probability generating functions and understand ...
3
votes
1answer
16 views

Is there a simplified form of this expression?

I have the following expression: $$ S = \sum_{k=1}^{K} \left(p_k \prod_{n=1}^{k-1}(1-p_n)\right) $$ All the $p_k$ are between 0 and 1. From numerical evaluations, I can see that when the $p_k$ are ...
1
vote
1answer
24 views

Why is the expected value of $|X|^p$ equal to $p\int_{0}^{\infty}y^{p-1}\mathbb{P}(|X|>y) dy$?

I'm trying to understand a passage from the book: A Basic Course in Probability Theory, Rabi Bhattacharya Edward C. Waymire, in the page 21. The calculation is the following: If $X$ is a random ...
1
vote
1answer
29 views

Why is this expectation finite?

For a random vector $X$, if $E (|c \cdot X|) < \infty$ for any vector $c$, then how can we show $ E(\|X \|) < \infty$? Thanks. Note: $\cdot$ means inner product.
-1
votes
0answers
11 views

probability tennis games, player winning all games [on hold]

a tennis game has 12 total players of equal competence as in the same exact level, and each of these players plays each of the other players exactly once, Find the probability that one of the 12 ...
1
vote
1answer
16 views

Elementary Probability: Expected Value

I must say, first, that this question IS a homework assignment and I do not wish an answer here, for I already posssess it. I want to know if there is a general procedure of simplification in this ...
1
vote
1answer
25 views

Probability about die

I have a die, I roll it $9$ times, what is the probability I get all six numbers at least once? I tried solving this by finding the probability that I don't get all numbers at least once, then ...
-4
votes
0answers
15 views

Poisson distribution question regarding colds [on hold]

The number of colds per year of an individual can be described by the Poisson distribution. Suppose that a new drug reduces the average number of colds per year from $3$ to $1$ and works for $75$ ...
0
votes
1answer
10 views

Possible conditional probability problem

I'm trying to understand the following question: Consider that a consumer buys with probability p, only one of three items A, B, C. If it buys, he will choose A, B or C of equiprobable manner. ...
0
votes
1answer
12 views

Boundary conditions for random walk

Consider a simple asymmetric random walk $S_n$ which goes up with probability $p$ and down with $1-p$. For $b<x<a$ let $$r(x) = P( S_n\text{ hits }a \text{ before }b |S_0 = x). $$ This ...
2
votes
2answers
68 views

Probability a product of $n$ randomly chosen numbers from 1-9 is divisible by 10.

I'm working on a problem where each number is chosen randomly from 1-9. Given $n$ numbers chosen in this manner, we multiply all of these together. I'm looking for the probability that this product is ...
0
votes
0answers
8 views

Weak convergence, measurability, uniform integrability

I've been facing the following problems: a) Let $ X_n \rightarrow X $ and $f $ be a measurable, bounded function. Prove that $ \mathbb{E}f(X_n) \rightarrow \mathbb{E}f(X) $ b) Prove that $ X_n ...
-2
votes
3answers
23 views

For Z ~ $N(0,1)$ and $X = Z^2$, find $f_X(x)$ of $X$. [on hold]

The question is pretty straight forward. I think I am not connecting something. $$Z \sim N(0,1)$$ Let $X = Z^2$ and find $f_X(x)$.
0
votes
0answers
7 views

Deriving a simple PDF

I am looking for deriving the pdf of $Z$ where $Z= (\sum\limits_{i=1}^N a_i X_i +Y_1)^2 + (\sum\limits_{i=1}^N b_i X_i +Y_2)^2$, where $X_i$ and $Y_i$ are independent, zero mean Gaussian random ...
0
votes
3answers
25 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 ...
0
votes
0answers
9 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
18 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
1answer
30 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
18 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 ...
0
votes
2answers
28 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
8 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
14 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 ...
-1
votes
0answers
14 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 ...
3
votes
2answers
30 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 ...
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
1answer
18 views

Drawing numbers with replacement

is the answer 1 - (1/3)^5 because Bob winning is 1/3 ^5
1
vote
0answers
26 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 ...
1
vote
1answer
18 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 ...
0
votes
1answer
51 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 ...
0
votes
1answer
32 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 ...
0
votes
0answers
21 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
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
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 ?
2
votes
2answers
29 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
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
25 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 ...
0
votes
0answers
43 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
1answer
19 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
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 ...
1
vote
1answer
22 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 ...