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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0
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3answers
45 views

Inequality for binomial distribution

In a bombing attack there is $50\%$ chance that any bomb will hit the target. Two direct hits are required to destroy the target completely. How many bombs must be dropped to give a $99\%$ chance or ...
0
votes
1answer
43 views

Tilde notation, what does it mean

A bit of a newbie question, but what exactly does tilde mean in this case? $$P(X>x) ∼ Cx^{−α}$$ This is in a context of some distribution (say Gaussian) that has Paretian tails. I have seen this ...
0
votes
1answer
33 views

Conf. Interval for a simple mixture of Normals

Imagine there's a prob. p=0.5 of choosing one machine or the other to take some measurements $X$ for an experiment. One machine ($N(\theta,10)$) is much less precise than the other($N(\theta,0.1)$) ...
2
votes
0answers
32 views

Weighted War - Game of Mind and Probability

Weighted War is a game of bidding, where: Both players have cards valued from $1$ to $11$ in their hands There is a third pile of cards from $1$ to $11$ face down on the table and shuffled,...
0
votes
1answer
22 views

Query on how the Probability is derived in the below scenario

A large pool of adults earning their first driver’s license includes 50% low-risk drivers,30% moderate-risk drivers, and 20% high-risk drivers. Because these drivers have no prior driving record, an ...
3
votes
1answer
32 views

Why am I under-counting when calculating the probability of a full house?

I was trying to answer this question. Find the probability of getting a full house from a $52$ card deck. That is, find the probability of picking a pair of cards with the same rank (face value), ...
1
vote
4answers
51 views

Probability: 5 cards drawn at random from a well-shuffled pack of 52 cards [closed]

A poker hand consists of 5 cards drawn at random from a well-shuffled pack of 52 cards. Then, the probability that a poker hand consists of a pair and a triple of equal face values (for example, 2 ...
2
votes
1answer
28 views

Problem of joint density function of random variable $(X,Y)$.

The joint density function of $(X,Y)$ is $$f(x,y)=\begin{cases}6(1-x), & 0<y<x,0<x<1\\ 0, & \text{otherwise}\end{cases}$$ Then which of the following are correct? $X$ and $Y$ ...
0
votes
2answers
26 views

Modified random one dimensional walk problem

If I take one step right from my initial position, I will fall off a cliff. My goal is to get to safety, which is exactly $x$ steps to my left. I decide which direction I will make my next step by ...
0
votes
0answers
30 views

Explanation of “When two functions $a(x)$ and $b(x)$ are not correlated on domain $X$, they can be separately integrated”

Hy everyone, I was reading this paper https://hal.inria.fr/hal-00942452v1/document , and I came up with a statement that I don't fully understand, nor could I found any info on it, so decided to ask ...
2
votes
1answer
42 views

PDE boundary condition question regarding limits

Just as a bit of background, I'm working with the Black-Scholes PDE and I'm testing some things out by taking an initial condition for it as $\sin(S/50)$, where $S$ is the spot price (but that's ...
0
votes
1answer
27 views

Why these two distributions are different when I calculated them?

Two insurers provide bids on an insurance policy to a large company. The bids must be between 2000 and 2200 . The company decides to accept the lower bid if the two bids differ by 20 or more. ...
-6
votes
1answer
68 views

Asymptotics of $\lim\inf X_n$, missing part of exam question [closed]

So I was working through an old exam and encountered the hilarious situation that part of the statement of the exercise was illegible. I was wondering if anyone could figure it out for me, so that I ...
4
votes
1answer
45 views

Bound for sum of normal distributions

I have encountered an exercise that was quite puzzling for me. Maybe someone can help me out here? So let $(X_n)_n $ be $N(-a,1)$ distributed, independent random variables where $a>0$. I need to ...
0
votes
0answers
12 views

Maximum-a-posteriori estimation with Gamma prior and scale-invariant likelihood

Let $\mathbf{X}$ be a vector of parameters with prior distribution $X_i \sim \text{Gamma}(\alpha, \beta)$ i.i.d. for $i = 1, \ldots, n$. Let us denote this prior by $p(\mathbf{X})$. We get to observe ...
1
vote
2answers
26 views

What is the distribution of the square of a sum of correlated Gaussian random variables?

Suppose a random vector $X\in\mathbb{R}^n$ follows a centered multivariate Gaussian distribution with zero mean and covariance $\Sigma$. We know that a linear combination of every elements in $X$ ...
0
votes
0answers
28 views

How to convert this text into probabilities?

How do we convert the table into probabilities? Let $S=\text{patient survives}$, $J=\text{patient is junior}$, $Sr=\text{patient is senior}$. Can we say that $P(S|J,T_2)=0.5$, or $P(S|Sr,T_2)=0.35$? ...
1
vote
1answer
40 views

Implications of conditional independence

Suppose two $\sigma$-algebra's $\mathcal{F}_1,\mathcal{F}_2$ are conditionally independent given some $\sigma$-algebra $\mathcal{G}$ i.e. for any $A\in \mathcal{F}_1$, and $B\in\mathcal{F}_2$, we have ...
10
votes
3answers
595 views

Calculating probability of winning best-of-7-games tournament. Why is my method wrong?

The question is as follows: A and B participate in a tournament of "best of 7 games". It is equally likely that either A wins the game or B wins the game, or the game ends in a draw. What is the ...
0
votes
0answers
13 views

error term in birth-death process

This is from notes: I have three questions I understand that for small $h$, when $h\to 0$, $h^2, h^3,h^4....$ are negligibly small compared to $h$, so I think the euqation 12 should be $(\...
0
votes
1answer
16 views

Predict value of random variable B if value of random variable A and correlation is known

Intuitively it is clear that if I have two variables $A, B \sim \mathcal{N}(\mu, \sigma)$ and I have $\rho_{A,B}=-1$, then if some sample $a$ of $A$ is $x$, then some sample $b$ of $B$ is $\mu-x$ ...
0
votes
2answers
65 views

Understanding an answer, finding the probabity of $x\in[0,1]$ being less than $\frac13$

Let $E$ be “the choosing of a number on the interval $[0, 1]$ such that the number is less than $1/3$. In this case, the use of the definition is a bit more subtle but essentially it gives $$\mathbb ...
0
votes
2answers
36 views

Probability in a inequality

I've been practicing probability for a while and now I encountered a math problem which I don't know how to approach. One integer is selected randomly from the set $[1,50]$ . What is the ...
-1
votes
0answers
21 views

Bernoulli trials: interrupting N and resuming it later. Is it cumulative or a new set?

I was wondering if I wanted to do N trials, lets say N=200000. But after tossing the coin 50000 times, I stop, and resume later the remaining 150000. Would that count as 200000, or as two different ...
0
votes
2answers
28 views

Probabilities in infinite Bernoulli type of series

While I was trying to solve the 1st problem from here I run into the following problem: find the probability of the events such as $1122213$ or $2122111116$ in infinite series of dice rolls which end ...
3
votes
1answer
67 views

Problem 48 in A First Course in Probability

I have an issue with problem 48 Chapter 2, page 51 in Sheldon Ross' A First Course in Probability (9th edition). The problem is as follows, Given 20 people, what is the probability that among the 12 ...
0
votes
0answers
34 views

Is $(X,Z)$ independent of $(Y,Z)$ if $X,Y,Z$ are mutually independent continuous r.v's? [closed]

Let $X$, $Y$ and $Z$ be mutually independent continuous random variables defined on a probability space. Is the random vector $(X,Z)$ independent of $(Y,Z)$ or not?
2
votes
1answer
43 views

Sample proportion and the Central Limit Theorem

Suppose that $ (\Omega,\Sigma,\mathsf{P}) $ is a probability space and that $ (X_{k})_{k \in \mathbb{N}} $ is a sequence of i.i.d. Bernoulli trials on $ (\Omega,\Sigma,\mathsf{P}) $, each with ...
2
votes
3answers
33 views

Which outcome will occur first [explanation]

Suppose I have a 100 sided die: 24 sides have the letter A 3 sides have the letter B What is the percentage likelihood to roll at least 6 As before 1 B. I believe the answer is (24/27) ^6 = 0....
0
votes
0answers
18 views

Number of prime exponents for Generalized Mersenne Primes

Please help me on the following scenario(s): Estimate the number of primes $p$ less than or equal to $x$ such that there is a prime of the form ${(a+1)}^p$ $-$ $a^p$ for all $a$ < $50$? What is ...
3
votes
0answers
27 views

Representation of point process

Let $E$ be a polish space and $N(E)$ be the space of finite integer value measures. It is known that for every $\mu \in N(E)$ exists $x_1, \dots, x_n \in E$ such that $$\mu = \sum_{i=1}^n \delta_{x_i}$...
-1
votes
1answer
26 views

Question from probability [closed]

10 students appeared for two examinations. 6 students passed the first exam, 5 students passed the second exam, and 3 students passed both the first and the second exam. What is the probability that ...
1
vote
1answer
34 views

One Independent Trial, What is the Chance of One Result Occurring First?

Each trial can result in one of three possible results. Result A happens 50% of the time. Result B happens 30% of the time. Result C happens 20% of the time. Each trial is independent. I want to ...
0
votes
1answer
23 views

Convolution mixture of a probability generating function (population genetics)

I'm trying to work through an old population genetics paper (see here). The following model assumes an infinite number of nucleotide sites and no recombination between different sequences (so you can ...
0
votes
1answer
43 views

Conditional expectation of a product of random variables

I have two independent continuous random variables $X$ and $Y$ with pdf's : $f(x)$ and $f(y)$ cdf's : $F(x)$ and $F(y)$ a constant $a$ I am trying to express using the given pdf/cdf functions the ...
1
vote
1answer
23 views

How to adjust estimation of probability according to new information

Suppose $\{a_1,a_2,\dots,a_n\}$ is a permutation of $\{1,2,\dots,n\}$. The probability of $a_i=j$ is estimated to be $p_{ij}$. The probability matrix might look like this $$ P=\left( \begin{matrix} ...
0
votes
2answers
70 views

Four people are rolling a die once. What is the probability of $2$ people getting the same number?

I am doing as following- $1^{st}$ people get a number from any $6$ number in $6$ ways, $2^{nd}$ people get a different number from rest of $5$ in $5$ ways, $3^{rd}$ people get another different ...
1
vote
1answer
37 views

How to prove this equation of probability?

The question is as follows. Of three independent events, the chance that only the first occurs is $a$, the chance that only the second occurs is $b$, and the chance that only the third occurs is $...
1
vote
0answers
26 views

What is the intuition behind right-continuous filtration?

I cannot understand the concept of it. So a filtration is right continuous if for every $t$ it holds that: $\mathcal{F_t}=\bigcap\limits_{\varepsilon>0}\mathcal{F_{t+\varepsilon}}$ But if for ...
0
votes
1answer
18 views

A fast randomized algorithm for the approximation of matrices

I am reading a paper which title is the same as the title of this question. Demonstration of Lemma 3.13 in the paper says that $ P\left ( \frac{\left \|Ax^{(j)} \right \|}{\left \| x^{(j)} \right \|...
1
vote
1answer
37 views

o(h) term in birth-death process

This is from the note I have two questions. Since $o(h)$ represents the probability of 2 birth and 1 death, 3 birth and 2 death, etc, why it still says $P(|X(t+h)-X(t)|>1)=o(h)$? shouldn't it ...
2
votes
2answers
63 views

How to calculate variance of W? Find the probability distribution of W?

$W=Y-X$ I have figured out that $E(W)=0.3$ by using this formula $E(X+Y)=E(X)+E(Y)$. I tried using the same formula with $E(X^2)$ and $E(Y^2)$ to find $E(W^2)$. I also tried using $V(X+Y)=V(X)+V(...
0
votes
1answer
40 views

Expectation of absolute random variables with mean 1 and standard deviation 1

For a random variable $\gamma \sim \mathcal{N}(\mu,\sigma)$ , were is $ \mathcal{N}$ is the normal distribution. What is the way to calculate the following: $ \mathbb{E}[|\gamma|] = ? $ And ...
1
vote
0answers
24 views

Probability distribution database

Is there some kind of probability distribution database? Very often I'm faced with a problem of fitting a distribution to data. And it also happens very often that exponential-looking data doesn't fit ...
5
votes
6answers
579 views

Find the probability of getting two sixes in $5$ throws of a die.

In an experiment, a fair die is rolled until two sixes are obtained in succession. What is the probability that the experiment will end in the fifth trial? My work: The probability of not getting ...
1
vote
0answers
26 views

Question on $\sigma -$algebra generated by 2 set.

Let consider the experiment "launched a coin three times". Describes the experiment space and give a filtration adapted to the problem. So if $H_i$ denote Heads at the i-th launched and $T_i$ denote ...
3
votes
1answer
16 views

Intuition of the relation between poisson process and order statistics

Lemma: Let $T_n$ be the time of the nth arrival in a Poisson process and $U_k$, $k=1,2....n$ be independent uniform on $(0,1)$. Then the order statistics of $U_1, U_2,....,U_n$ have the same ...
-1
votes
1answer
20 views

Integrating the product of pdf's [closed]

I have N independent Normal random variables~(0,$\sigma^2$). The resultant density function is a product of the individual densities. Now to integrate this product from $\lambda$ to $\infty$ i.e, $\...
1
vote
1answer
37 views

If I flip a single coin twice, what are ALL the possible events?

I understand that all the possible outcomes are: HH, HT, TH, and TT. The sample space is: S = {HH, HT, TH, TT} But what are the possible events? My textbook says that it is 2^n. So my understanding ...
1
vote
1answer
20 views

The problem of ksdensity plot in Matlab

I want to verify that random numbers generated by exprnd match the exponential distribution. I used Matlab's ksdensity function –...