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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1answer
36 views

Combining a set of conditional probabilities

I'm interested in combining a set of conditional probabilities into one. For example, if I know the following probabilities: P(illness|male) ...
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2answers
28 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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4answers
47 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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0answers
7 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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1answer
41 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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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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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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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 ...
4
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3answers
50 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
20 views

Basic conditional probability question

$\sum_{c}p(a|c)p(c|b)=p(a|b)$. Does this equation hold true? If it is true, how to prove it mathematically?
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4answers
339 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
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
265 views

Entropy of matrix vector product

Consider a random $n$ by $n$ circulant matrix $M$ whose first row entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first ...
0
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1answer
34 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
27 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
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 ...
2
votes
1answer
57 views

Chernoff-like bound for small intervals in tail distribution

I am searching for a Chernoff-like bound that controls the probability of small intervals in the tail distribution. More specifically, let $X_1, \ldots, X_n$ be independent random variables with ...
3
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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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0answers
23 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 ...
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2answers
3k views

Possible combinations for 20 character alphanumeric identifier

I need to know the total possible unique variations there can be on an identifier that is made up of 20 alphanumeric characters, where the characters are A to Z (all upper case), and the digits 0 to ...
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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
15 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 $X$ ...
1
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1answer
25 views

Expectation with respect to empirical distribution

Let $(\Omega,\mathcal{A})$ be a measure space and $X$ a random variable with distribution $P$. The expectation of some measurable function $g$ with respect to $P$ is $$ \mathbb{E}_P[g(X)] = ...
3
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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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1answer
53 views
+50

What is the probability that no two consecutive boxes have blue balls.?

There are red and blue balls which can be filled in 5 boxes.All balls are similar except color. what is the probability that no two consecutive boxes have blue balls. Assume:A ball can be either ...
2
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2answers
3k views

Probability of getting 'k' heads with 'n' coins

This is an interview question.( http://www.geeksforgeeks.org/directi-interview-set-1/) Given n biased coins, with each coin giving heads with probability Pi, find the probability that on tossing the ...
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0answers
17 views

Convolution of independent but 'different' probability distributions

I have the following two probability distributions they relate to a particular ice-cream: ...
1
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3answers
69 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
16 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 ...
0
votes
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
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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0answers
19 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 ...
0
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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
22 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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0answers
22 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 ...
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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 ...
2
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2answers
48 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 ...
0
votes
1answer
299 views

partial differentiation of function of expectation of random variable

We have $E(U)=\int_0^B V f(V) dV + B \int_B^\infty f(V)dV$; Here $V$ is random variable. $E(U)$ stands for expectation of $U$. We have $Z=f(E(U))$ i.e. $Z$ is function of $E(U)$. Can we write ...
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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 ...
1
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1answer
304 views

Theater row brainteaser

I bumped into this brainteaser The Theater Row: Eight elegible bachelors and seven beautiful models happen randomly to have purchased single seats in the same 15-seat row of a theater. On the ...
1
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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 ...
0
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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 ...
0
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2answers
21 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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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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0answers
18 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, ...
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 ...
7
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3answers
2k views

PDF of a sum of exponential random variables

Let $X_i$ for $i=1,2,...$ be a sequence of i.i.d exponential random variables with common parameter $\lambda$. Let $N$ be a geometric random variable with parameter $p$ that is independent of the ...
0
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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 ...