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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Bound for the variance of a stochastic process

Given a random variable $X$ and $N$ realizations of the stochastic process associated to $X$, a theorem gives a bound for the $\sigma^2[X]$: $$\sigma^2[X]\le\frac{1}{4}(A-a)$$ where $A$ and $a$ are ...
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120 views

Chebyshev Inequality

I am reading a research paper, and the author claims to get to a desired result by making use of the Chebyshev Inequality. I can get to the desired result also with some reasoning, but I fail to ...
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41 views

If a sequence of events $A_n$ converges to 0, then does the probability of the intersection equal 0?

Here's an elementary question which I believe is true, but my intuition about infinite intersections is not very solid. Let $A_n$ be a sequence of events such that $P(A_n) \rightarrow 0$ as $n \...
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27 views

Rearranging Question

Hi guys I have come to the following inequality but can't seem to workout how to get to my final step. I know it is just rearranging the equation but I haven't been able to get it or I would like ...
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3answers
823 views

Probability that a student knows the answer

The Probality that a student knows the correct answer to a multiple choice question is 2/3 . If the student does not know the answer , then the student guesses the answer . The probality of the ...
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18 views

Finding the probability of a probability density function

Suppose that $f(x) = e^{−x}$ for $0 < x$. find $P(1 < X)$ I know typically we integrate $f(x)$ from $1$ to $\infty$ but in this case $x = 1$ is not included, how do I go about doing this? All ...
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1answer
31 views

Reversing Combinations to find probabbilities of variable two state systems

First I'm going to lay out the problem and do an insanity check to make sure I've planned out properly so far. The first question is probably trivial, but the second one is definitely a bit more ...
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1answer
48 views

Probability help.

Did something in class the other day ad it made me curious on what the probability was. There were 19 people in class and each of us had to pick any 5 items out of a list of 12, order did not matter. ...
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1answer
40 views

The Distribution of a Function of a Distribution

Given a PDF of $X$: $f(x; \alpha) = k/x^\alpha$ , for $x\ge5$, and $0$ for all else. And $\ln(X/5)$ has an exponential distribution with parameter $\alpha-1$ My question is what exactly is $\ln(X/...
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62 views

On a problem of sphere-packing for Reed-Solomon codes

Suppose we have an $[n, k+1, n-k]$ Reed Solomon code $\mathcal C$ over $\mathbb F_q$, where $n-k=d$ is the minimum distance, and suppose that $d=2t+1$. We know that for every $r \in \mathbb F_q^n$ the ...
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1answer
678 views

Joint distribution of range $(R=X_n-X_1)$ and mid-range $(V=\frac{1}{2}(X_1+X_n)$order statistics

Let $X_1,X_2, · · · , X_n$ be independent and identically distributed Uniform random variables on the interval (0, a) for a > 0, each having a density function $f(x) = \frac{1}{a}$, $0<x<a$. Let ...
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0answers
77 views

Proof of Law of Iterated Expectation for discrete case

One of the Law of Iterated Expectations states that $$E\left( E\left( X\mid G \right)\mid H \right)=E \left( X\mid H \right),$$ where $H\subseteq G$. My "desperate attempt" of proof goes like this. $...
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1answer
30 views

Need help with contingency table for predicting signup result

I am trying to calculate the number of people in a sample of the population who fit certain criteria. For example, let's say we have a population of 1,000 people. We also know the following about the ...
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1answer
52 views

Joint Probability of Random Variables

Suppose I took measurements $\{X_i\}$, which are all independent and they follow a normal distribution $X_i\sim N(\mu,\sigma)$. I am asked for the joint probability of all of the measurements. Based ...
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1answer
121 views

Probability: Random Variables and Probability Distributions

1) The function: $F(x)=k(1-(1/2)^{[x]})$, $x > 0$ Is the distribution function for a discrete random variable X. Here, [x] denotes the integer part of x (i.e., the greatest integer less than or ...
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2answers
115 views

Bayes Theorem application confusion

The probability of a person NOT having Lyme disease is .99793 When tested for Lyme disease there is a false positive rate of .03 and a false negative rate of .063 .... find the probability that a ...
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1answer
74 views

Variance in offspring genotypes. Binomial distribution

Background Here is first some vocabulary: Diploid: phase in the life cycle where the individuals carry two chromosomes of each type, just like in humans (exception of the sexual chromosomes). ...
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2answers
93 views

Finding Distribution function from probability density function

Given $$f(x) = \begin{cases} x, \; 0 < x <1\\ 2-x, \; 1 \leq x < 2\\ 0 \text{ everywhere else} \end{cases}$$ as our P.D.F, I must find the corresponding distribution function. I know that $...
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1answer
41 views

A and B are independent events with $P(A)=.4$ and $P(B)=.5$, determine $P(A\cup B)$

If A and B are independent events and $P(A) = .4$ and $P(B) = .5$, determine $P(A\cup B)$. I did $P(A \cap B) = .4 + .5 = .9$ and since $P(A\cup B) = P(A) + P(B) - P(A\cap B)$ $P(A\cup B) = .4 + .5 -....
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2answers
237 views

conditional expected value - Poisson process plus random variable

I've struggled with this actuary excercise for a while and I don't know how to do it: Each claim can be characterized by two random variables $(T,D)$, where $T$ is the moment of reporting the claim ...
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3answers
5k views

Box containing Red and black balls

A box containing 4 red balls and 6 black balls . Three balls are selected randomly from the box one after the another, without replacement . The probality that the selected set has one red ball and ...
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2answers
2k views

Probability that 4 cards drawn from a deck of 52 cards are all hearts?

Draw 4 cards from a deck. Determine the probability that all of them are hearts. Will it be 1/13 * 1/12 * 1/11 * 1/10 ? I'm not sure, any help will be helpful, thanks!
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1answer
126 views

Exchangeable/Independent Bernoulli Distribution

Let P be a uniform random variable on the interval $(0,1)$ with density function f(p) = 1, $0<p<1$. Let $X_i|P$, i = 1,2,...,n be independent and identically distributed random variables having ...
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1answer
215 views

Travelling from one destination to another

This is the problem : Manish has to travel from A to D changing buses at stops B and C enroute. The maximum waiting time at either stop can be 8 minutes each, but any time of waiting up to 8 minutes ...
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1answer
33 views

Recurrence of states in a function of a Markov chain

Suppose $X$ is a Markov chain (or process, for that matter) and suppose further $f(X)$ is also a Markov chain. Let $s$ be a recurrent state in $X$. Is there a general way to determine the recurrence ...
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55 views

Finding the appropriate bounds of integration for this joint probability.

Let $f(x_1, x_2) =\begin{cases} 4x_1x_2, \ 0 < x_1 < 1, \ 0 < x_2 < 1 \\ \\ 0, \text{elsewhere} \end{cases}$, be the pdf of $X_1$ and $X_2$. Find $P(X_1 = X_2)$. I understand that ...
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2answers
31 views

Let $X,Y$ be random variables in uniform distribution, $0\leq X\leq 3$,$0\leq Y\leq 4$, the probability of $X\leq Y$

Let $X,Y$ be random variables in uniform distribution, $0\leq X\leq 3$,$0\leq Y\leq 4$, I want to compute the probability of $X\leq Y$ For each $X$, the probability of $X\leq Y$ is $\int_X^4\frac{1}...
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3answers
187 views

Sum of squares of two integers divisible by five [closed]

Supposing $x,y$ are natural numbers, what is the probability that the sum of their squares are divisible by 5? I am getting $1/3$ as squares can only end with $0,1,4,5,6,9$. So $36$ pairs are ...
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1answer
155 views

Proof that random walk visits zero infinitely many times

Since the Green function $G(x,1)=\sum\limits_{n\in \mathbb{N}_0}P(S_n=x), x\in\mathbb{Z}^d$ gives the expected number of visits to $x$ in a random walk, I'm asked to prove the following: I have to ...
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0answers
140 views

Definition and derivation of conditional expectation/probability

I read quite a few books introducing the notion of conditional probabilities/expectation by putting a formula out there coming from what they call "intuition". Can someone provide me a good measure ...
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2answers
620 views

Stationary distribution of random walk on a graph

If you do a random walk on an undirected, connected graph, is the stationary distribution for the probability that you have just traversed edge $e$ uniform over all edges no matter what the graph ...
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1answer
45 views

Proving equidistribution

We have three stochastic variables. $X\sim L(1)$, $Y_{1}\sim Exp(1)$ and $Y_{2}\sim Exp(1)$. $Y_{1}$ and $Y_{2}$ are independent and equidistributed. Note that the $L$ stands for the continuous ...
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35 views

Pattern Recognition terms a little fuzzy

I've been trying to learn more about probability and the websites I have visited are not describing the relationship to each other that well (when to use what, for what purpose in conjunction with ...
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2answers
422 views

How many hands are there with exactly 5 hearts after drawing 7 cards from a deck? [closed]

Draw 7 cards from a deck of 52 cards. How many hands are there with exactly 5 hearts? Will it be something like $$\frac{1!}{(52!51!50!49!48!)\cdot(7!6!5!4!3!)}$$ I'm pretty sure its wrong, any help ...
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3answers
533 views

At a hospital's emergency room, patients are classified and 20% of them are critical, 30% are serious, and 50% are stable.

At a hospital's emergency room, patients are classified and 20% of them are critical, 30% are serious, and 50% are stable. Of the critical ones, 30% die; of the serious, 10% die; and of the stable, 1%...
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1answer
161 views

Expected value of a min of a function of a random variable and a constant

I am reading a paper on contracts and there was an expected value calculation which got me confused. Consider the following primitives $\delta \in (0,1)$, an exogenous parameter $D$ a positive ...
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2answers
55 views

A probability problem I am trying

Suppose $X_i$ are iid Bernoulli with parameter $p$, $0<p<1$. and $N$ is Poisson $\lambda$, and $N$ is independent of $X_i$. Let $S_N=X_1+X_2+\dots+X_N$. Find the distribution function and hence ...
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3answers
332 views

probability 5 people have birthdays in 2 months

The Question: What is the probability that in a family of 5, all birthdays will be in just two different months? What is wrong with the logic (12*11*2*2*2)/(12^5) (12 choices for the first person, ...
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1answer
58 views

Number of ways of distributing 7 distinct balls into 7 distinct boxes with exactly one box with 3 balls.

This is what I tried. I can distribute the balls in four ways: 1) 3,1,1,1,1 2) 3,2,1,1 3) 3,2,2 4) 3,4 For 1) I can first pick 5 boxes in ${7 \choose 5}$ ways and then pick 3 balls in ${7 \choose ...
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1answer
590 views

# of seating arrangement in a 6 seat car

I'm getting hung up on a probability question: A car has six seats including the driver’s, which must be occupied by a driver. In how many ways is it possible to seat 4 people if all 4 can drive. for ...
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1answer
254 views

Calculate the Entropy Change if 5 Previously Tossed Dice Are Turned to All “1”

Relevant Equations: S = Boltzmann*ln(W) where S is entropy and W is the number of microstates. I have thought about this two ways. 1 way. Look at each die separately. Let macrostate 1 = number of ...
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1answer
28 views

How to set up this type of probability question

(Problem) A fishing boat has 10 worms and 10 leaches as bait. The bait is chosen at random. Find the probability that the 5th worm is drawn as the 6th draw from the container. Assume that there is ...
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1answer
442 views

Hacking with the last four of your SSN

Suppose a hacker gets a hold of the last four numbers of your social security number (the serial number). What is the probability that the hacker randomly guesses your full social security number? ...
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1answer
142 views

Mutliplication rule vs. complement principle

If you roll four dice, what is the probability that there will be at least two dice showing the same number? I suppose the answer is 1 - [(6*5*4*3)/6^4] using the complement principle. But would how ...
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1answer
268 views

If a Rubik’s cube is tossed 6 times what is the probability that the cube will land on the red side at least 4 times?

A Rubik’s cube in which each side is painted one of six colors (white, orange, red, blue, green and yellow). Suppose each side of the Rubik’s cube consists of only one color, if the Rubik’s cube is ...
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1answer
60 views

prove: if B $\subset A$, then Probability(B) $\le$ Probability(A)

B $\subset A$, then P(B) then P(B) $\le$ P(A) I've seen a video to this answer and some yahoo answers to this question but it's still not clear to me how you derive the answer.
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1answer
78 views

Probability mass function - find the second tallest boy in a room.

A kindergarten class consists of $12$ boys and $4$ girls. The children are arranged from tallest to shortest. Assume that all $16!$ rankings are equally likely, and no two children are exactly the ...
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1answer
432 views

Rubiks Cube Landing on Red

A Rubik’s cube in which each side is painted one of six colors (white, orange, red, blue, green and yellow). Suppose each side of the Rubik’s cube consists of only one color, if the Rubik’s cube is ...
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1answer
69 views

Apparently same probability questions with different answers.

I was reading A first course in probability by Sheldon Ross when and then I came up with this question. This is how he introduces the famous problem of points Independent trails, resulting in a ...