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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3 views

Two players toss a coin.

Two players $A$ and $B$ toss a coin. A has a coin $C_A$, B has a coin $C_B$. Probability of tail for $C_A = 1-a$, of head: $C_A = a$ Similary for $C_B$. Now, they are tossing on turn. The A starts. ...
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0answers
3 views

Does this sequence converge? If yes, what is the limit?

Assume $\{k_n\}_{n\geq 0}$ a sequence of natural numbers such that $k_0=0$, $k_n\leq k_{n+1}\leq k_n+1$, and $\lim_{n\rightarrow\infty} \frac{k_n}{n}=\alpha\in(0,1)$. So $\{k_n\}$ is an ...
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1answer
14 views

Variance of a random variable between 0 and c.

My professor says we need to know how to solve a problem like this for our upcoming exam and I can't find anything in my textbook or notes related to this at all. Can anybody help make this ...
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1answer
9 views

Find probability of event

Task is: Find probability of 4 aces laying in row in a deck of 36 cards. All possible shufflings of 36 deck is $36!$ I can place 4 cards in a row with $33$ different ways. And each way can be $4!$ ...
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2answers
10 views

Probability density function of random variable $X-Y$

Suppose $X$ and $Y$ are independent random variables. $X$ and $Y$ are continuous and given by exponential and uniform density functions. Find the probability density function of the random variable $X ...
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2answers
1k 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 ...
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1answer
9 views

Assume that P(B) > 0. Prove that if P(A1|B) < P(A1) then P(Ai | B) > P(Ai) for some i = 2, … , k.

Suppose k events from a partition of the sample space Ω, i.e., they are disjoint and ∪ i=1 to k over Ai = Ω. Assume that P(B) > 0. Prove that if P(A1|B) < P(A1) then P(Ai | B) > P(Ai) for some i = ...
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0answers
5 views

Linear transforms of Normal dist

If $X_t = \sqrt{t} Z$ where $Z \sim \text{N}(0,1)$ Then show the distribution of $X_t - X_s$ for $s<t$ Just wanted to check, would this be $\sim \text{N}(0,t-s)$ or $\sim \text{N}(0,(t-s)^2)$ ?
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0answers
18 views

Guessing a password from a list of N passwords

I want to know if my solution to the problem is correct: I am given a list of n passwords to enter an account and only one will grant me access to it. I pick one and I test it. If it's incorrect, I ...
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0answers
27 views

Another fun card game involving probability

Two people, (call them C and D), decide to play a card game for fun. They use an ordinary fair deck of $52$ cards, well shuffled, and randomly draw cards from it one a time without replacement, both ...
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1answer
8 views

Asymptotic stopping time for a ball-drawing problem

Take two different boxes, one with $N$ red balls and one with $N$ blue balls. Remove balls one at a time from either box with equal probability. When only one color is left, the (expected value of ...
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1answer
7 views

Construction of Probability Generating Function in Branching Process?

So I'm trying to construct a probability generating function for the following scenario: 1/5 of a rabbit population does not reproduce. 4/5 have 3 offspring each, and the probability of male or ...
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0answers
13 views

probability of collision approximation,matlab,maple

I'm working in the field of communication network. I have this equation p=((1-(1-(((2.*(1-2.p))./(1-p-p.(2.p).^m)).(1./Wi))).^(n-1))) I want to solve for p, but I failed(maple and matlab), can ...
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2answers
37 views

Clever way of finding $\int_0^\infty x\Phi(x)\phi(x)dx$

Suppose that $\Phi$ and $\phi$ are the Standard Normal c.d.f and p.d.f. respectively. Then, evaluate $$\int_0^\infty x\Phi(x)\phi(x)dx$$ There is no use of my trying to show my approach because ...
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2answers
12 views

Two processes. Expected value.

On a computer running two processes $ X_1, X_2 $ at the same moment. $ X_1, X_2$ mean time work processes, respectively. $ X_1, X_2 $ have exponential distribution. $$ E (X_1) = E (X_2) = 60s.$$ ...
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1answer
25 views

Counting and Probability String Length

Consider strings that can be made up from the set $\{a, b, c, d, e, f, \cdots, z, 0, 1, 2, \cdots, 9\}$ 1) How many strings of length 8 contain either the letter '$x$' or '$1$'? 2) What is the ...
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1answer
38 views

Calculating probabilities for complex random variables

I am having some trouble understanding/formulating how one computes probabilites given a (somehow complex) continuous random variable. For example, if I define a random variable $Z$ as: ...
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0answers
12 views

Probability of profit

I came across the following question in a book:- $Q.$ Cards are drawn one by one at random from a well shuffled pack of $52$ cards. $(a)$Find the probability that exactly $n$ cards are drawn before ...
2
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0answers
30 views

An inequality for symmetric random walk

I need to show that if $(X_j)$ are symmetric i.i.d. random variables with partial sums $S_n:= \sum_{j=1}^n X_j$, then for all $x \geq 0$ $$P(|S_n| > x) \geq \frac{1}{2} P(\max_{1 \leq j \leq n} ...
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0answers
8 views

Probabilities for the repetition of the same experiment $N$ times

Sometimes one experiment we want to discuss in terms of probabilities is in truth the same as performing another experiment $N$ times. I have a doubt on how to relate the probabilities for the ...
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0answers
15 views

Decision-making with random term

Consider the following situation. There are multiple options to choose from based on an attribute related to those options. For example: ...
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2answers
30 views

What is the expected number of people who are shorter than both of their immediate neighbors?

A total of n people randomly take their seats around a circular table with n chairs. No two people have the same height. What is the expected number of people who are shorter than both of their ...
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2answers
42 views

Find the distribution - coin is tossed three times

A fair coin is tossed three times. Let $X$ be the number of heads that turn up on the first two tosses and $Y$ the number of heads that turn up on the third toss. Give the distribution of $X$, $Y$, $X ...
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0answers
26 views

What is the probability of recovering from 0 − 40

A game in tennis consists of a sequence of points played with the same player serving. A game is won by the first player to have won at least four points in total and at least two points more than the ...
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2answers
20 views

Probability and limit - proof of equality

Could anyone explain why this equality is true? Is there some intermediate step that could be used to prove it? If I were to guess, I'd guess it's certainly equal, but guessing is not enough I'm ...
2
votes
2answers
29 views

Showing that supremum function is integrable

Let $g_1(\omega),g_2(\omega),...$ be integrable functions defined on $\Omega$ with $g_n\rightarrow g$ and $g$ is integrable and also $\lim \int g_n=\int g$ . Define $h(\omega)= \sup_n g_n(\omega)$. ...
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1answer
21 views

Show that $\Omega\setminus A_1, Ω\setminus A_2,\ldots, \Omega\setminus A_n$ are independent

Let $(\Omega, \Sigma, P)$ be a probability space and let $A_1, A_2, \ldots , A_n$ be independent events in this probability space. Show that $\Omega\setminus A_1, \Omega \setminus A_2, \ldots , \Omega ...
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votes
1answer
36 views

Finding Variance

I am a little confused on how to go about finding different parts of the Variance of a random variable. Here is the question. A total of $n$ balls, numbered $1,.. n$, are put into $n$ urns, also ...
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0answers
11 views

Conditional Distribution of Ordered Statistics [on hold]

Let $X_1,...,X_n$ be the order statistics of a set of $n$ independent uniform (0,1) random variables. Find the conditional distribution of $X_n$ given that $X_1=s_1,...,X_{n-1}=s_{n-1}$. I honestly ...
3
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2answers
39 views

Find the probability that $X$ is even when following a geometric distribution [on hold]

Suppose that $p \in (0, 1)$ and that $X$ is a discrete random variable with a geometric distribution with parameter $p$. Find the probability that $X$ is even.
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0answers
24 views

Help with two probability questions. Classic definition of probability.

The first can be done using condition probability, but was wondering how to do it just with the classic definition of probability? Both questions are in the same part of the book, and therefore i ...
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1answer
28 views

Find distribution and the expected value of final grade [on hold]

A performance is graded independently by three experts (the possible grades are as follows: 1, 2, 3, 4, 5), and then the highest and the lowest mark are crossed out. The remaininggrade is the final ...
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1answer
8 views

Normal Distribution: Statistics

I'm having a lot of trouble trying to remember the formulas on how to calculate these questions. Any help would be great. An automobile insurer has found that repair claims are Normally distributed ...
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0answers
23 views

Probability for large nose and/or large ears

John has got a large nose and Mary big ears. Mary gives birth to their 5 children. Each one of them inherits the big ears with a probability of 0.5 and the large nose with 0.5 as well.Each child ...
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1answer
31 views

Jee Main 2015 Question. Probabilty

If $12$ identical balls are to be placed in $3$ identical boxes, then the probability that one of the boxes contains exactly $3$ balls is: (1) $22 \times(\frac{1}{3})^{11}$ (2) $\frac{55}{3} \times ...
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0answers
9 views

Determining the expected number of Grim Patrons after a bouncing blade on a board with no minion cap [on hold]

The title of this question refers to the card game Hearthstone and a particular card interaction. See http://hearthstone.gamepedia.com/Grim_Patron http://hearthstone.gamepedia.com/Bouncing_Blade So, ...
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0answers
9 views

Normally distributed random variable

Could you please answer this question with an explanation, I am new in this subject.
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0answers
12 views

conditional expectation uniqueness

Conditional expectation is unique up to a set of probability measure zero, but if $Z=E[X|Y]$ and $Z_2$ almost surely equals $Z$, then is $Z_2=E[X|Y]$ still the case? I think this is false but can't ...
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1answer
13 views

Prove that a symmetric distribution has zero skewness

Prove that a symmetric distribution has zero skewness. Okay so the question states : First prove that a distribution symmetric about a point a, has mean a. I found an answer on how to prove this ...
2
votes
1answer
29 views

Application Birkhoff ergodic theorem

Let $(X,\mathcal{B},m,T)$ be a probability preserving transformation. Let \begin{align*} I:&=\{f\in L^1: f=f\circ T\}\\ B:&=\{g-g\circ T: g\in L^1\} \end{align*} I have to show that $$ ...
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2answers
57 views

Expected value of the larger of two claims [on hold]

Claim amounts for wind damage to insured homes are independent random variables with common density function $f(x) = \frac{3}{x^4}$ for $x > 1$; and $f(x)=0$ otherwise. Suppose two claims are ...
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0answers
9 views

Find lower bound of probability value using Chebyshev's inequality

Given density function of random variabel $X$ is $f(x) = 3x^2$, for $0 \lt x \lt 1$. Use Chebyshev's inequality to find lower bound of probability value : $P(5/8 \lt x \lt 7/8)$ $P(1/2 \lt x \lt 1)$ ...
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3answers
30 views

Problem on Baye's formula

I was reading A First Course in Probability by Sheldon Ross. I think I quite understood the below problem but I still feel fuzzy. Problem: In answering on a multiple choice test, a student either ...
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0answers
203 views

Is independence a transitive property?

If the events $A$ and $B$ are independent and the events $B$ and $C$ are independent, does this necessarily mean events $A$ and $C$ are independent? I used coin tosses to try to model this with $A = ...
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1answer
46 views

Sequence of letters of length $5$

How many sequences of letters are there of length $5$ with exactly $2$ vowels? Don't count "y" as a vowel. Pretty lost on this one. I know it involves a $\binom{5}{2}$ part, but I feel like that's ...
1
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1answer
115 views

Combinatorial Marble Choosing

A bag contains $3$ red marbles, $3$ green ones, $1$ lavender one, $6$ yellows, and $4$ orange marbles. How many sets of five marbles include either the lavender one or exactly one yellow one but not ...
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1answer
138 views

Calculating the variance, mean, and autocorrelation of a time series.

How can I calculate the mean, variance, and autocorrelation function: $$Y_t=5+Z_t+ 0.6Z_t-1$$
3
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0answers
24 views

How to calculate the distribution within a series

The probability of event $A$ happening is $50.7\%$. The probability of event $B$ happening is $49.3\%$. What is the probability that in a series of $100$ trials, there will be at least one point ...
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0answers
16 views

Is this an algebra of events?

An algebra of events is a non-empty $F$ family of subsets of the sample space $\Omega$ closed under the union and the complement. That's to say $F \subset P(\Omega)$(power set) that verifies: 1st) $F ...
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0answers
18 views

Interchanging infinite double sum and expectation

Let $\xi_i$ be a sequence of independent and identically distributed standard normal random variables and consider sequences $\{b_i\}$ and $\{c_j\}$ such that $\sum_i b_i<\infty$ and $\sum_j ...