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 views

Joint distribution probabilities

I have a question that is similar to the following(made up here): The construction of a tower of cards is done is two stages, procrastination and the actual building. The time in minutes needed to ...
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1answer
32 views

Roll Dice- Expected Winnings

I have a problem like this: At a charity game you pay \$1 to roll a die. If you roll a 6, you get \$5. Otherwise, you get nothing. How do I set up a probability distribution and what is the ...
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2answers
12 views

Probability of multiple choice

Suppose there are two multiple choice questions with 4 choices each. Assume you answer the first question by choosing one of the four answer uniformly at random. You answer the second questions by ...
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0answers
20 views

Is the martingale propertey preserved by taking weak$^*$-limits?

Let $(\Omega,\mathcal F)$ be a measurable space, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that ...
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2answers
17 views

Normal Distribution finding values

The question says: X is normal with mean -1 and variance 4. Find the value $x_0$ for which the probability is $.2676$ that $X$ will take on a value less than $x_0$. I know this has to deal with ...
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1answer
14 views

Probability of same last four digits of a telephone number

Hi all this is really my first post here .. Yesterday I was talking to a girl and asked her for her phone number . Once she gave it to me we realized that we got exact same last four digits . Hence ...
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3answers
10 views

conditional Probability proof with inequalities

Let A, B be events taken from a sample space Ω (with Pr(A) > 0 and Pr(B) > 0). If Pr(B|A) < Pr(B), prove that Pr(A|B) < Pr(A). I am a bit confused with this one. Any help would be appreciated. ...
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1answer
24 views

Russian Roulette and conditional probability

Let's say you play Russian roulette with a 6-chamber gun and there is only one bullet in it. Your friend spins and pulls the trigger, he's still alive, and then he gives the gun to you and you need to ...
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1answer
21 views

Repeatedly rolling an $n$-sided die

Suppose I roll an $n$-sided die once. Now you repeatedly roll the die until you roll a number at least as large as I rolled. What is the expected number of rolls you have to make? I know the answer ...
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1answer
9 views

Probability of 2 of three independent events occuring

Three objects are thrown at a target. The probabilities the individual objects will connect with the target is .75, .85 and .90. Find the probability that at LEAST two of the objects hit the target? ...
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1answer
13 views

Probability: arithmetic on Random Variables

I have a question about the arithmetic on random variable in probability. Question: Are the events $\{X=Y\}$, $\{Y=Z\}$,$\{Z=X\}$ independent? My solution: $$ P(X=Y,Y=Z,Z=X) = {(0.5^2 ...
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1answer
12 views

Probability of Billy being the first triplet called to the desk

There are 11 students in a class, three of whom a triplet -- Billy, Annika, and Catherine. The teacher calls all 11 students in random order one at a time to his desk, but Billy is in the bathroom ...
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1answer
29 views

Find the conditional expectation of N given that there were exactly 2 heads in the first 3 tosses.

We have two biased coins. The first one yields heads with probability 0.1 and the second one yields heads with probability 0.9. We choose one of the two coins randomly (with probability 0.5 each; we ...
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3answers
1k views

Finding the probability of soccer game's results

It's a little basic question. I'm so much into soccer and I love how statistics and football can be integrated. But I have little knowledge about statistics, yet I do have information about the ...
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0answers
10 views

The Birthday Problem Generalization to multiple types of varying probability

This is the Generalization to multiple types as given in Wikipedia. But where I am stumped is how to treat those type if they have varying constraints. Such as, in a group of men and women what is ...
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0answers
10 views

Waiting time probability question

I want to solve the following problem: A dentist works 4 hours a day. Patients arrive on the average of 1 per 20 minutes and one patient spends on average 15 minutes with the dentist. Both time ...
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0answers
7 views

Marginalisation and conditional probability

I'm afraid is an extremely simple question, but I didn't understood completely why if both $O_1$ and $O_2$ can be marginalised over $R$ the following holds: $ P(O_2 = o_i | O_1 = o_j) = \sum_r P(O_2 ...
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1answer
39 views

Is there a name for the trivial probability distribution P(X=x) = 1 for a unique x?

Is there a name for the trivial probability distribution given by $P(X=x) = 1$ for a unique $x$ and $P(X=y) = 0$ for all $y \ne x$? I know it is very trivial, but since it is the distribution that ...
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2answers
85 views

Show that $ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$ is unbiased estimators of $θ$.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the uniform distribution on the interval $(θ, θ + 1)$. Let $$ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$$ Show that $\hat{\theta}_2$ is ...
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0answers
13 views

Inequality similar to Hoeffding

I have a coin with heads probability $p_1$. I toss it $n_1$ times. Let $\hat{p}_1$ be the empirical heads probability. Then we know from Hoeffding that $$P\left( \left|\hat{p}_1-p_1 \right| \geq ...
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1answer
325 views

Invariance of the correlation coefficient under linear transformations

Show that for arbitrary random variables $X$ and $Y$, and constants $a,b,c,d$ with $a$ and $c$ nonzero, $$ \mathrm{Corr}(aX+b,cY+d) = \begin{cases} \mathrm{Corr}(X,Y)\quad&\text{if }a \text{ and ...
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1answer
17 views

proof of property of exponential distribution, using taylor polynomial

I want to prove that if we have an exponential distribution with parameter $\lambda$, we have that $P(X \le x)=\lambda x + o(x)$. I want to do this by using Taylor-series and the lagrange remainder ...
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3answers
18 views

Sigma Algebra Measurable R.V

I am trying to figure out what random variables are measurable with respect to sigma algebra given by $[1-4^{-n}, 1]$ where $n= 0, 1, 2, ....$ if $[0,1]$ is the sample space. I believe I can do with ...
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1answer
77 views

Are “most” continuous functions also differentiable?

Let $A$ be a nonempty open subset of $\mathbb{R}$. Consider a function $f : A \rightarrow \mathbb{R}$. Given that $f$ is continuous, what is the probability that it is differentiable? I suspect it ...
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1answer
18 views

Different answers to this probability problem

The Grunters and the Screamers are playing for the Grand Championship, which is a best of 7 series. The first team to win 4 games wins the Championship. Each team has a $\frac{1}{2}$ probability of ...
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1answer
20 views

Poisson process and moment generating function

If we have a Poisson process $ \lbrace N(t), t > 0 \rbrace $ with rate $\lambda > 0 $ and if we have a random variable $S$ having a uniform distribution on the interval $[0,2]$. I was wondering ...
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1answer
22 views

Odds of winning more than 50% of many bet of different %

I made a bet with a friend and I would like to know if I'm ahead or not. We have a package of 6 games. Each game have a different probability of a team to win. If it's a tie 3-3, it's a push. If it's ...
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0answers
28 views

What does it mean “choose one randomly”?

I do not get what this sentence mean. I have a set of numbers $\mathcal{L_{\ell}}=\{n_1, n_2, \dotsc, n_{\ell}\}$. Choose randomly one number from $\mathcal{L_{\ell}}$. Does it mean the probability ...
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1answer
23 views

Distribution of Chi-Square divided by its degrees of freedom?

I have the following: $$\frac{2n}{\chi^2_{\nu=2n}}$$ Does this simplify to be a $\chi^2_{\nu=1}$ distribution by any chance? Or is there a rule to get rid of the $2n$? Any help would be ...
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1answer
21 views

Probability of item distribution with a restriction

I'm having a hard time analyzing my research data, and was wondering if anyone had any suggestions? I've reworded the question so it is presented more like a statistics problem. There are $x$ number ...
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4answers
111 views

What does it mean when it says “probability of death doubles” every 8 years? Can probability of death exceed 136%?

I've seen on a lot of websites that your probability of death "doubles" every 8 years. However, the way they calculate the probability of death seems to lead to counterintuitive conclusions. e.g on ...
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0answers
10 views

Construct SDE with two uncorrelated Brownian motions

Using $Y(t) = wX_1(t) + \sqrt{1-w^2}X_2(t)$ as a model to construct a process where X1 and X2 are brownian motions with drifts and brownian increments $dX_1(t)= \mu_1dt + \sigma_1dW_1(t)$ ...
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3answers
38 views

Fair rolling for a six sided die

One die is rolled with the following payoffs: $$\begin{array}{c|r} 1 & \$25 \\ 2 & \$5 \\ 3 & \$0 \\ 4 & -\$10 \\ 5 & -\$10 \\ 6 & -\$15 \end{array}$$ How much would I need to ...
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0answers
13 views

Probability for Wiener process [on hold]

How to calculate such probability $$\mathbb{P}(\max \limits_{0\le \tau \le t} W(\tau) \ge x)$$ where W is the Wiener process.
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1answer
37 views

Establishing fairness in test grading

Consider a group of 75 students who sit an exam consisting of 20 open questions, and are then randomly divided into 3 groups of 25 students {A, B, C} for grading by 3 different persons. Let us ...
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0answers
6 views

Differentiability of random processes.

I know the appropriate criterions for mean-square differentiability of random processes. These criterions are connected with covariance function of a process. Are there any criterions for ...
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2answers
16 views

Probability Question - team draw from field of 32

For a sport tournament where two-man teams are drawn from a sample of 32 without replacement, what is the probability of two men being on the same team one year and then two years in a row?
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0answers
24 views

Average minimum distance

I posted a question earlier here and someone pointed out that it might not be possible to find a closed form solution due to the elements of $\mathbf{g}$ and $\mathbf{f}$ defined below coming from a ...
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0answers
5 views

“Reverse” distribution-tails

Chernoff, Markov and Chebyhev all give some upper bound for tail probabilities, e.g. Chebyshev gives us $Pr[|X-E[X]| \geq t] \leq \frac{Var[X]}{t^2}$. This is quite helpful, but what if I would ...
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1answer
23 views

Mean and variance: Gaussian is the most conservative assumption

"given only the mean and variance of a distribution, the most conservative assumption that can be made about the distribution is that it is a Gaussian having the given mean and variance" I've read ...
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0answers
6 views

Singular distributions: Applications and Instances

This is the duplication of the question I asked here. I repeat it here with hope of getting new answers. Singular distributions are special mathematical objects. They have an interesting property ...
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1answer
34 views

Conditional Probability Problem

Consider a (hypothetical) state with four cities: $C_i$, $i=1, \cdots, 4$. The probability that a resident of the state lives in city $C_i$ is $p_i$ (and $\sum_i p_i = 1$). If a resident lives in city ...
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0answers
13 views

Branching processes, extinction probability

Why do we assume that a branching process starts with one ancestor. What happens to extinction probability if we have more than one ancestor in generation Y(0)?
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2answers
356 views

Taking seats on a plane: probability that the last two persons take their proper seats

100 men are getting on a plane (containing 100 chairs) one by one. Each one has a seat number but the first one forgot his number. So he randomly chooses a chair and sits on it. Others do know their ...
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1answer
21 views

arbitrage free price in martingale measures

Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ...
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1answer
27 views

Envelopes and Mailboxes

We suppose $n$ and $p$ are two positive integers. A) In how many ways can you divide $p$ identical envelopes in $n$ mailboxes? (Each mailbox can hold several envelopes at the same time) B) In how ...
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0answers
37 views

Bounds for sum of random variables

Let $A_1,...,A_M$ be random variables, not necessarily independent. For each one of them I know that $P( A_i \geq a )\leq B_i, \quad i=1,2,...,M$. How can I retrieve lower/upper bounds for ...
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1answer
66 views

What is the probability that two products are equal?

Consider a square $n$ by $n$ matrix $A$ and two vectors $v$, $w$ of dimension $n$. The entries of $A$, $v$ and $w$ are either $-1$ or $1$ with equal probability and are i.i.d. and all three are ...
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0answers
15 views

Borel-Cantelli lemma: convergence of series [on hold]

Consider a sequence $(X_{n} )$ of r.v.'s, $M_{n} = \max_{k \leq n} |X_k|$ and $u_n$ a nondecreasing sequence. In order to analyze the convergence of the series $\sum_n P(X_n > \varepsilon u_n)$, ...
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1answer
143 views
+50

How minimize $\sum p_b \ln{p_b}$?

I have a multiset $A = \{a_1,\dots,a_n\}$ of integers. Let $q = P(a_i = a_j)$ when $i$ and $j$ are chosen independently and uniformly from $\{1,\dots, n\}$. Let $B$ be the set of integers in $A$. We ...