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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Finding the distribution of the sum of n independent random variables having exponential distributions

My graduate level probability class asks us to calculate the distribution of the sum of n independent exponentially distributed random variables. I am trying to perform many convolutions but it gets ...
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1answer
18 views

How many ways can an integer $i$ appearing in a sequence with multiplicty at least $j$, be minimal

Let us construct an integer sequence of length $n$, where the integers are chosen from $\{1, 2, ..., k\}$, with i.i.d. uniform probability $\frac{1}{k}$. I want to compute the probability ($p_{ij}$) ...
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2answers
35 views

Help needed to solve probability problem

I am trying to solve the following problem. A fisherman is equally likely to go fishing at one of the three ponds A,B,C. The probability to catch fish if he cast his rod at pond A is 0.4 , at pond B ...
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0answers
8 views

In an incomplete market does every payoff function admit at least two arbitrage-free pricing?

Consider an arbitrage-free (not necessarily complete) market. Prove or disprove the following assertion. If the market is incomplete, then every payoff function $A : \Omega \rightarrow \mathbb{R}$ ...
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1answer
47 views

How to remember these probability results?

If $A,B$ and $C$ are $3$ events, then $P$(Exactly one of $A,B,C$ occurs)$=P(A)+P(B)+P(C)-2[P(A \cap B)+P(B \cap C)+P(A \cap C)]+3P(A \cap B \cap C)$ $P$(Exactly two of $A,B,C$ occur)$=P(A \cap ...
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1answer
21 views

Basic conditional probability question [on hold]

$\sum_{c}p(a|c)p(c|b)=p(a|b)$. Does this equation hold true? If it is true, how to prove it mathematically?
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4answers
49 views

What is the probability of drawing 3 balls such that none of them is red?

Given a bag containing $8\ \color{red}{red}$ balls and $4\ \color{green}{green}$ balls, what is the probability of drawing $3$ balls at random such that $\mathbf {none}$ of them are ...
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1answer
32 views

Expected % of heads flipping coins of different odds

So this is an analogy for a real world example but for simplicity. So if I were to flip a normal coin ten times I would expect heads 50% of the time or 5 head results. I could then compare this to the ...
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4answers
582 views

Minesweeper probability

I ran into the situation pictured in the minesweeper game below. Note that the picture is only a small section of the entire board. Note: The bottom right 1 is the bottom right corner tile of the ...
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1answer
36 views

Probability that distance of two random points within a sphere is less than a constant

I am looking for help with the following problem: Two points are chosen at random within a sphere of radius r. How to calculate the probability that the distance of these two points is < d ? My ...
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0answers
8 views

Sampling of a changing mixture model

Let f, g, and h be probability density functions, and X, Y and Z be random variables respectively following f, g and h. The mixture model: ${\text h = \frac{f + g}{2}}$ states that the distribution ...
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3answers
51 views

Doubt about a probability excercise

I'm a statistics teacher at a college. One day a student came with a doubt about an exercise about probability. The text goes like this: A person has two boxes $A$ and $B$. In the first one has ...
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1answer
15 views

Random Permutation Polynomial With Fixed Inputs

Assume we pick uniformly random a permutation polynomial, $T$, of degree one. we define all polynomials over $\mathbb{Z}_P$. We have fixed inputs $x_i$ (e.g. $x_i \in [1,100]$) My Question: Is ...
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0answers
24 views

Calculating Combinations / Permutations [on hold]

How do I calculate the number of outcomes as a whole of a series of individual tests with there own outcomes? For example, the best description I could think of would be: There are 10 tests and each ...
1
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1answer
19 views

Double integral proving that a function is a probability density

If $$g(x,y)=f(x+y)/(x+y)$$ for $x,y>0$ and $$\int_0^{\infty} f(z) \, dz = 1$$ How do you show that $$\int_0^{\infty} \int_0^{\infty} \frac{f(x+y)}{x+y} dx \, dy = 1$$ as well?
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0answers
17 views

Uniqueness of the transformation turning random variables into IID uniform

We have two random variable $X:\Omega \to \mathbb R $ and $Y: \Omega \to \mathbb R^d, d \in \mathbb N$, $F_Y$ is the density function of $Y$ and $F_{X|Y=y}$ is a regular density function of $X$ ...
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0answers
29 views

Properties of independence and conditional independence

Recently, I see some properties from conditional independence wiki page https://en.wikipedia.org/wiki/Conditional_independence I don't quite understand the properties of "Rules of conditional ...
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0answers
19 views

Convolution of independent but 'different' probability distributions

I have the following two probability distributions they relate to a particular ice-cream: ...
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0answers
17 views

Compute distribution in Hidden Markov models

Let $Z_1, Z_2, ..., Z_n$ be the latent variables, and $X_1, X_2, ... X_n$ be the observed ones in a hidden markov models. Let's assume that the parameters of the hidden Markov models are known: the ...
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2answers
45 views

Can someone give me real world example of uniform distribution [0,1] of a continuous random variable.

Can someone give me real world example of uniform distribution [0,1] of a continuous random variable, because I could not make out one.
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0answers
19 views

what is the intuition behind the SRSWOR formula?

I earlier asked about Slovin's Formula, and learned shortly thereafter that it was derived from this formula. $n=\dfrac{n_0}{1+\dfrac{n_0}{N}}$, Where $n_0=\dfrac{z^2p(1-p)}{e^2}$. So, breaking it ...
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0answers
18 views

Distributions of identical and distinct objects [on hold]

I'm having an issue figuring this problem out. I'm not sure how I should go about it exactly, all I know is that it needs to divided into stages, each its with its own set of cases. So here's is ...
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3answers
70 views

How to calculate the limit of $(\frac{x}{x+1})^x$

I am looking at the probability of losing $x$ games in a row, in a game where the probability of winning is $1/x$. (For example, if this is a fair casino game, what is the probability of losing $x$ ...
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0answers
20 views

Probability of me being one of a group

I've heard that in a certain country with a population of about 140,000,000, 200 people become missing on a daily basis. If I want to calculate the random probability that one of those people would ...
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0answers
17 views

what is formula to this eqution [(256)16]1/32+[(169)6]1/12 [on hold]

how to solve this equation [(256)16]1/32+[(169)6]1/12 what is formula of this? What is the closed form expression for this? What is the right domain for this Hamiltonian 2? what is the right ...
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votes
1answer
21 views

Conditional Probability Equivalence

I am reading this paper here: http://www.utm.utoronto.ca/~weisber3/articles/SobervBJPS3.SP.pdf which claims on page 10 that $p(E \wedge [H_x \wedge \sim R]) + p(E \wedge [H_x \wedge E])= p(H_x \wedge ...
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2answers
24 views

Convergence Events with States

Ignatz repeatedly rolls a fair $6$-sided die. What is the probability that he rolls his first $5$ before he rolls his second (not necessarily distinct) even number? I don't know what to do about the ...
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1answer
23 views

3 Events, Conditional Independence

Given $A,B,C$ such that: $$ P(A\mid B),P(A\mid B^c),P(B\mid C),P(B\mid C^c) \text{ are known } $$ and that $A,C$ are conditionally independent given $B$, so that: $$ P(A\mid B\cap C)=P(A\mid ...
2
votes
2answers
49 views

How do mathematician make sense of “outcome” and “events” in probability?

One of the biggest challenge for me to understand probability is to make sense of this concept of outcomes and events. To put it plainly, it just doesn't feel like mathematics anymore when we talk ...
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3answers
48 views

Intuitive meaning of the probability density function at a point

I understand how to integrate probability density functions to find probability within a certain range. However, what I don't understand is what it would mean to set the variable (say x or y) to a ...
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votes
1answer
34 views

What time should Celia aim for in her sixth race to make the team? [on hold]

To be on the 1-km race team, Celia must have a mean time less than 5 min 50 sec in her 6 tryout races. Her times in 5 races are: 6 min 2 sec, 5 min 53 sec, 5 min 45 sec, 6 min, and 5 min 34 sec. What ...
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1answer
21 views

When randomly distributing n points amongst m people, what are the odds that one certain person will get a certain amount of points?

I'm mostly curious about how to find this in general, but the actual problem is with 20 points and 5 people. I know probability problems are very counterintuitive, and thus I was unsure after ...
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1answer
26 views

Probability with $n$ successes before $m$ failures

Independent trials resulting in a success with probability $p$ and a failure with probability $1 − p$ are performed. What is the probability that $n$ successes occur before $m$ failures? Given ...
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2answers
22 views

Mathematical Backing for Observations seen in Adding Independent Random Variables Together

So I have a function Y = F(N) that takes as an argument an integer number N and returns a summation of N sine-waves of different random parameters. I have plotted the results of two function calls ...
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0answers
19 views

Probability, expected frequency and resultant distribution skewed or not?

A population consisting of a certain proportion of defective items has mean $\mu = 2$. If a sample of 4 items is examined and repeated 200 times, obtain a) probability of an item being defective, ...
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2answers
27 views

defective component and probability

An electronics industry uses three plants from A, B and C in the ratio of $3$ to $2$ to $1$. However $1\%$ of components manufactured by A, $2\%$ of components manufactured by B and $3\%$ of ...
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2answers
26 views

Games and statistics

Three individuals A, B and C alternate in contention of a game according to the following rules: A plays with B and the winner plays with C. The game continues until one of the individuals to win two ...
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0answers
16 views

Coupon collector problem with $k$ distinct coupon sets to complete

In the standard coupon collector problem we have an urn with $n$ different coupons, from which coupons are being collected, equally likely, with replacement. Simple analysis shows that the expected ...
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1answer
21 views

Finding conditional probability using Baytes formula

Suppose that chips for an integrated circuit are tested and that the probability that they are detected if they are defective is $0.95$, and the probability that they are declared sound if in fact ...
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0answers
26 views

Maximum difference between tails in absolute value

I toss a fair coin $n$ times. Some notation: $S_i=$ difference between #heads and #number of tails after the first $i$ tosses, $1\leq i\leq n$. $M_n=\max(S_1,S_2,\dots,S_n)$, ...
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1answer
47 views

Probability that AJ is guilty

A crime has been committed by a solitary individual, who left some DNA at the scene of the crime. Forensic scientists who studied the recovered DNA noted that only five strands could be identified and ...
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0answers
5 views

Concentration bounds on Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...
2
votes
1answer
45 views

Error in Billingsley?

Problem 8.25 in the third edition of Probability and Measure by Billingsley (1995, p. 142) is as follows: Suppose that an irreducible [Markov] chain of period $t>1$ has a stationary ...
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2answers
39 views

Probability that among 3 random digits two different one

I have been trying to solve the following problem: What is the probability that among 3 random digits, there appear exactly 2 different ones? The formula for no repititions is: ...
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1answer
45 views

Age distribution when meeting

I have a question regarding Poisson process. I will tell the story in the context of a player-monster game. Consider a player who is born at $t=0$. He will win the game if he can survive until ...
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votes
2answers
36 views

Selecting k distinct numbers from an array with increasing probability distribution

I have to select k distinct numbers from an array such that probability of a number getting selected is more if it is at the end of the array (probability increases linearly). I'm thinking of ...
1
vote
1answer
30 views

Chessboard pawns arrangement clarification

I have a 8 X 8 chessboard, and 8 identical pawns. These pawns are arranged at random. What is the probability that the pawns are arranged in such a way that each row and column have only one pawn? My ...
3
votes
1answer
49 views

rolling a single die ten times

I have the following problem on a homework assignment for my Probability theory course: You roll a single six sided die ten times. What is the probability that you roll four 1's, three 2's, and three ...
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0answers
28 views

Conditional distribution

One point is chosen at random in the square $Q=\{|x| + |y| \leq 1\}$. Let $(X, Y)$ coordinates that point. a) The random variable $X$ and $Y$ are independent ? b) Find the density of $X$ given that ...
0
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2answers
26 views

Simplifying with Summation

This is a problem out of my statistics book but my issue is simplifying from Step 3 to Step 4 below: Step 1: var X=$\sum\:p_i\:(x_i-E[X])^2$ Step 2: var X=$\sum\:p_i[x_i^2+E[X]^2-2x_iE[X]]$ Step ...