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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-2
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1answer
24 views

probabilities of the given events [on hold]

A pair of dice is rolled. Find the probabilities of the given events. (Enter exact numbers as integers, fractions, or decimals.) (a) The sum is 4. (b) The sum is 4, given that the sum is less than 6. ...
3
votes
1answer
43 views

Largest age difference between great-great-…-grandparents?

If I go up $n$ generations, with some assumptions, I will have exactly $2^n$ ancestors of different ages. For simplicity, I'll call these ancestors my $n$-parents. Let $A_n$ be the difference between ...
0
votes
0answers
26 views

Data analysis: How did people beat the Great Hall game?

This is the game: There is a Great Hall with 102 doors. 100 of these doors lead to one of 100 different side rooms. The 101st door, at the end of the Great Hall leads to the Great Tower, where ...
0
votes
1answer
28 views

Ways to arrange ALGEBRA so AA occurs

So the permutations of this qould be 7!, and I know that there are 2 objects of type A, but how can we isolate the events where those objects occur consecutively?
0
votes
1answer
24 views

Exit time of simple random walk on $[-a,b]$

It can be proved using martingales and the optional stopping theorem that the expected exit time of a random walk on $(-a,b)$ beginning at $0$ with $a,b>0$ is $ab$. How can this be shown using a ...
0
votes
2answers
51 views

Random Variables and Probability Distributions

Little Help here Q-For a laboratory assignment, if the equipment is working, the density function of observed outcome X is f(x) = 2(1-x), 0 < x< 1 otherwise 0 ...
3
votes
2answers
474 views

What is the expected number of suits in a hand of 4 cards?

To find the expected number of suits the formula is $E(Num Suits) = 1*P(1 Suit) + 2*P(2 Suit) + 3*P(3 Suit) + 4*P(4 Suit)$ For the probability of getting 4 suits I got ${13 \choose 1}^4 {4 \choose ...
1
vote
0answers
29 views

Flip a coin, then repeat an experiment n times. Show exchangeable but not independent

We flip a fair coin. If it is heads then we roll a die n times, if it is tails we sample a number n times from the set {1, 2, 3, 4} with replacement. We denote the resulting n numbers by X1, ..., Xn. ...
2
votes
1answer
39 views

Follow-on question to “Fifty men and thirty woman…”

This questions relates to this question Fifty men and thirty woman are lined up at random. How do I find the expected number of men who have a woman standing next to them. and the answer given by ...
1
vote
1answer
48 views

There are 8 balls which appear identical. However, 1 is heavier than the rest. How do you find the ball with 2 weighings?

I understand there are similar problems but I am not sure how to go about constructing this problem with set of balls that are not exponents of 3^n. I know I need at least 2 weighings to find the ...
1
vote
1answer
16 views

Prove that for any kΕ[0,1] the interval is a confidence interval [on hold]

I have to prove that for any kε[0,1] the interval is a confidence interval
0
votes
2answers
33 views

Finding an unbiased estimator for a parameter, dicrete variable

Let $X : \Omega \to \mathbb{N}$ be a random variable. Define $p_i = P(X=i), \ \ i \in \mathbb{N}$. Find an unbiased and consistent estimator for $p_1$. I need to find an estimator $\alpha_n(X_1 + ...
1
vote
1answer
46 views

Probability of same birthday

I think I solved this problem but I would like to know if I am right or wrong, I am not quite sure. We assume that the year has 365 days and the birthdays are uniformly distributed. We want to find ...
0
votes
1answer
22 views

What is the variance of sixes that appear when a 1-6 die is rolled 1000 times?

So I'm having a bit of trouble wrapping my head around variances... I calculated $\mathbf{E}[X]$ as $\frac{1000}{6}$, but that's the easy part. To calculate the variance, I'm trying to calculate the ...
3
votes
0answers
29 views

Estimating/approximating a very high dimensional unbounded poisson's equation

Consider the poisson equation on an unbounded domain. Suppose that the solution is known to exist. $$ \Delta u=f $$ I would like to estimate the solution of the this equation at a given point $x_0$. ...
1
vote
3answers
51 views

Expected number of cards drawn before drawing a $4$ or $5$

I'm working on the following problem: Compute the number of expected cards drawn from a standard 52 card deck (without replacement) until a $4$ or $5$ is drawn. I tried to model it using a ...
0
votes
1answer
16 views

Probability formula for a specific case

My university's exam period is 30 days long. Say I have five exams, each with an equal likelihood of being on a certain day within the period. Then for the $d$th day of the period, what is the ...
1
vote
0answers
22 views

The convergence of probability for $X_nY_n$ and $X_n/Y_n$

Suppose that $X_n, Yn$ ($Y_n\neq 0$ a.s) converge to $X,Y$,respectively, in probability. I need to show 1) $X_nY_n \rightarrow XY$ in probability. 2) $X_n/Y_n \rightarrow X/Y$ in probability. My ...
1
vote
1answer
15 views

Martingale roulette system

I'm making a roulette system simulator, specifically right now the Martingale roulette system. So what I do know about the system that there is an Anti-Martingale too, which is the same, but you have ...
3
votes
0answers
34 views

A strange Jensen's inequality for function of two variables

I am reading a paper where they use implicitly the following "Jensen's inequality which i find quite strange. Moreover i did not find this result in any textbook so, i would like an opinion before i ...
0
votes
1answer
26 views

probability question, two uniformly distributed independent events, what's the probability that a third event will occur? [duplicate]

let's say there's two events, a and b both a and b are uniformly distributed and have a range of [100,400] a and b are independent i know that the probability that a=A is 1/300 and the probability ...
0
votes
2answers
19 views

What is the probability that out of a deck of 16 cards that you will be dealt 2 cards with the same number?

Suppose you are playing with a set of 16 cards, which consists of 4 cards of each color (red, green, blue, and yellow) with each colored card having a different number on it (1, 2, 3, or 4). In other ...
0
votes
2answers
29 views

How many ways are there to distribute 6 distinguishable objects into 4 indistinguishable boxes so that each of the boxes contain at least 1 object?

How many ways are there to distribute 6 distinguishable objects into 4 indistinguishable boxes so that each of the boxes contain at least 1 object? Can anyone tell me how should I approach this ...
1
vote
1answer
30 views

Conditional probability - Basketball player

A basketball player has a chance of 80% on his free throw shoots and 30% on his three-point shoots. If he makes free throw shoots N times and then makes three-point shoots as many times as the ...
0
votes
1answer
18 views

Equivalent of random variable sequences in distribution?

Suppose that $X_n, Y_n$ are sequences of random variable on probability space $\Omega$. If $Xn,Yn$ converges to $X$ ( some random variable ) in distribution, then is $X_n=Y_n$ almost everywhere (a.s)? ...
0
votes
1answer
20 views

Frequency Distribution and Throughput

I am conducting an experiment on a couple of computer systems but the results I have don't make sense to me. I made each system perform 1000 operations: System A performs operations at a rate of ...
1
vote
1answer
21 views

Setting up an expected value problem involving a Poisson distribution

I have the following problem: Let $N$ be the number of losses and let $$ T[N] = \begin{cases} \$0 & \text{ if }N = 0\\ \$10,000 & \text{ if }N = 1 \\ \$17,500 & ...
0
votes
2answers
16 views

If X is an exponential random variable with parameter lambda and x>0…

If $X$ is an exponential random variable with parameter $\lambda$, and $c>0$, show $cX$ is exponential with parameter $\frac{\lambda}{c}$. I don't remember being taught this in class, nor does ...
1
vote
1answer
18 views

Characteristic function of $\sum_{t=1}^N a_t X_t$ given certain independence conditions

Let $S=\sum_{t=1}^N a_tX_t$ where each $a_t$ is Bernoulli with probability $\frac{1}{2}$ for $1$ and also $\frac{1}{2}$ for $0$. Moreover, it is also given that the vector $(a_1,\ldots,a_N)$ is ...
0
votes
0answers
21 views

inclusion–exclusion principle formula in probability for general case

For the system reliability analysis of complex systems, I'll use the formula below after a point in a custom computer code. I know that for the n=2 case, formula result is $P_1+P_2-P_1.P_2$ ($P$ : ...
1
vote
3answers
55 views

Probability of dying

I have this scenario: 10 people with 100% of probability of dying in 1 month. So I assume that at the end of the month I'm gonna have: 10 dead people; 0 alive people. I also have this ...
3
votes
1answer
32 views

Prove $E[XE[Y\mid\mathcal{G}]] = E[YE[X\mid\mathcal{G}]]$

Show that for bounded $X$ and $Y$ that $E[XE[Y\mid\mathcal{G}]] = E[YE[X\mid\mathcal{G}]]$. Attempt: Suppose that $X = _{\mathcal{X}_F}$, where $F \in \mathcal{D}$. Then for every $B \in ...
1
vote
0answers
26 views

How many ways can 5 different jobs be assigned to 3 different employees so that each employee is assigned at least 1 job?

Solution given made use of the inclusion and exclusion principle: Number of ways = 3^5 - (3 x 2^5) + 3 I don't quite get the solution :/ Can't we just assign each of the employee one job and then ...
0
votes
1answer
45 views

Conditional Probability : How to get $P(Y_{1} \mid S_{1} ,Y_{2} \mid S_{2}) = P((Y_{1},Y_{2}) \mid (S_{1},S_{2}))$

Given $S_1$,the probability that $Y_1$ and $S_1$ are matching is $P(Y_1\mid S_1)$ Given $S_2$ ,the probability that $Y_2$ and $S_2$ are matching is $P(Y_2 \mid S_2)$ Also , whether $Y_1$ and ...
0
votes
2answers
18 views

How many ways can n books be placed on k distinguishable shelves if no 2 books are the same and the positions of the books on the shelves matter?

This is the solution provided: C(k+n-1,n) x n! I know that C(k+n-1,n) denotes the number of ways n indistinguishable books can be placed on k distinguishable shelves. But I'm not too sure how ...
0
votes
0answers
10 views

Expectation Maximization question

I came across this question while practicing EM question but I don't understand how to apply EM in this scenario. What's the latent variable here? Is it the grade of each student? What will be the ...
0
votes
0answers
11 views

How many ways are there to distribute 12 distinguishable balls into 6 distinguishable bins so that 2 balls are placed in each bin?

This is what I've done: Number of ways = C(12,2) x C(10,2) x C(8,2) x C(6,2) x C(4,2) x C(2,2) x 6! So basically what I'm trying to do over here is to pick 2 balls randomly for each of the bins. I ...
0
votes
1answer
36 views

An inequality regarding expectation of random variables

Let $X,Y$ be positive-valued, well-behaved random variables. Further, let $g(\cdot) \ge 0$ and $f(\cdot)\ge 0$ be two functions and $E(\cdot)$ denotes expectation operator. I am trying to prove the ...
4
votes
1answer
27 views

Expected number of turns for SPROUT

As a mathematical father (and with apparently plenty of time on my hands) I long ago computed the expected number of turns for a number of children's games that are effectively Markov maps. (Chutes ...
3
votes
1answer
31 views

Bivariate Probability question

Let $X$ be the time (in minutes) that John spends waiting for a bus on his way home from uni, and let$ Y$ be the time he spends waiting for a train. $X$ and $ Y$ have joint density ...
-6
votes
0answers
30 views

The probability that Z is between 0 and -1.61 [on hold]

What is the probability that Z is between 0 and -1.61? Would be good if you could show working. Thanks.
0
votes
0answers
23 views

If there are two different stationary distributions, then there are infinitely many distributions in reducible markov chain

If there are two stationary distributions μ1 and μ2 there are actually infinitely many stationary distributions: (pμ1 + (1 − p)μ2) is also a stationary distribution for any real number 0 ≤ p ≤ 1. How ...
-1
votes
1answer
44 views

There are 20 red marbles, 10 blue marbles, and 5 white marbles in a jar.

There are 20 red marbles, 10 blue marbles, and 5 white marbles in a jar. Select a marble without looking, note the color, and then replace the marble in the jar. We’re interested in the number of ...
3
votes
1answer
94 views

Rectangle randomly thrown on chessboard

) I'm an electrical engineer and having a tough problem with... math :) geometry and probability... Here's the problem : We have an infinite chessboard. Each square of the chessboard is of known ...
-2
votes
1answer
37 views

Equation involving conditional probabilities and independent events [on hold]

Given S1,the probability that Y1 and S1 are matching is P(Y1|S1). Given S2,the probability that Y2 and S2 are matching is P(Y2|S2). whether Y1 and S1 are matching is independent with whether Y2 and ...
0
votes
1answer
20 views

What's the interpretation of this random variable

Let $(\Omega,\mathscr{F},P)$ be a probability space and $X$ be a random variable that takes values in $\mathbb{N}$. Define $$q(n)\equiv P(X=n)\quad n\in\mathbb{N}$.$$ So $q$ is just the probability ...
0
votes
2answers
23 views

Probability of a deck of cards [on hold]

In a deck of cards, what is the probability of drawing four kings? Not sure if it is rule of compliment?
-1
votes
4answers
20 views

probability of bad watch [on hold]

a watch store has 10 watches; one of them does not work. If we pick 3 watches by random, what is the probability that the bad watch in in our sample?
0
votes
0answers
11 views

Need a lower bound for a discrete monotonic distribution

I'm staring at the following expression: $$ \displaystyle \frac{\sum_{i=0}^{n}\sigma_i\left(\sigma_i-\sigma_{i-1}\right) w_i}{\sum_{i=0}^{n} \sigma_i^2}$$ I need to come up with a lower bound to ...
-2
votes
0answers
30 views

Probability distribution function and probability density function [on hold]

In constructing the bridge shown below, an engineer is concerned with forces acting on the end supports caused by a randomly applied concentrated load P, the term ‘randomly applied’ meaning that the ...