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
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1answer
23 views

Paradoxical Game Show Problem

Here's a problem that has had me scratching my head for a long time: Imagine you're in a game show, and are presented with 2 boxes. You are told that both boxes contain a sum of cash, but one of the ...
0
votes
0answers
9 views

Probability in the urn model without replacement.

In an urn with $p$ total marbles, $p_A$ are white and $p-p_A$ are black, we know that the probability of drawing at least $m_A$ white marbles out of a $m$ without replacement follows the cumulative ...
0
votes
2answers
11 views

Normal Distribution,standard deviation and probability question.

According a study, the duration of a match in World Cup is approximate normally distributed with the mean 111 minutes and standard deviation 5 minutes (including the break between the halves). ...
0
votes
1answer
21 views

Odds betting - how to ensure the next stake will return all previous losses and itself

This problem is quite hard to explain so I will demonstrate with an example. Say I stake $£1$ on a horse (odds are irrelevant at this point) to win a particular race. The horse loses, so I decide to ...
0
votes
1answer
20 views

Expected value of the floor function of a sum of two variables

In a recently published paper I have encountered the following equality. Let $U$ be a random variable uniformly distributed in $[0,1]$ and let $Z$ be a Gaussian variable with mean zero and standard ...
2
votes
1answer
33 views

Example of non continuous random variable with continuous CDF

Can someone provide an example of $X$ being a non-continuous random variable with continuous cumulative distribution function? For instance: $X$ is discrete if it takes (at most) a countable number ...
1
vote
1answer
30 views

How to prove that doing something earlier in time increases the probability of success?

Let's say I have two events A and B. If I do A before B I will succeed. If I do A after B I will fail. Now, the problem is that I don't know when B happens. Intuitively I can say that if do A as ...
0
votes
0answers
20 views

What is the probability that in an estate of 10,000 computers ( PCs, laptops etc) that work optimally for up to 4 years?

I have 10,000 computers ( PCs,laptops etc) in my organsiation. I only have the funding to replace 500 computers a year ( this varies year on year depending on capital allocations). If I am only able ...
-2
votes
0answers
26 views

Birthday Problem [on hold]

This is an extension of birthday problem, please help In a class of 85 students, let X be the number of students who share a birthday with at least two other members of the class. a) ...
-1
votes
2answers
22 views

Geometric distribution related probability questions [on hold]

I am learning Probability, and I have this problem. Suppose $X\sim {\cal Geom}(p_1)$ on $\{1,2,3,...\}$, $Y\sim {\cal Geom}(p_2)$ on $\{1,2,3,...\}$, and $X, Y$ are independent. Let $S=X+Y$. ...
0
votes
0answers
14 views

Inference Two Sample

During the regular season of the NBA, the great Larry Bird made 3,960 free throws out of 4,471 attempts for a career free throw percentage of 88.6%. As of March 21/11, British Columbia’s own Steve ...
0
votes
1answer
27 views

Conditional expectation of function of random variable [on hold]

Show E[u(Y)|Y]=u(Y) given that u(Y) is a real function.
0
votes
1answer
16 views

Probability with two uniformly distributed cost

I have two uniformly distributed costs that are random and statistically independent. They are 3,000 to 6,000 and 3,000 to 12,000. I am trying to find the probability that the total cost of these two ...
0
votes
2answers
18 views

Variance of a dice roll

I am currently working on a problem and am unsure if I approached it correctly. Here it is: ...
0
votes
2answers
9 views

Determining level of significance when hypothesis is an interval

For a normally distributed sample: σ = 60 Sample size = n = 12 Sample mean = x = 3450 Null Hypothesis = H_o ≠ 3500 Hypothesis = H_1 = 3500 I need to determine the smallest significance level at ...
0
votes
1answer
15 views

how to find the maximum of the cross-entropy of a discrete random variable?

For a discrete random variable $x$, the cross entropy is $$H(x) = -(p_1\log p_1+\cdots+p_n\log p_n)$$ , so what is the maximum of $H(x)$? Here is what I tried, I compute the gradient as follows ...
0
votes
2answers
41 views

Probability notation P versus Pr

I have come across both $P(…)$ and $Pr(…)$ being used to represent probabilities. Is there any difference in the meaning of these notations, or are they just different shorthands? I seem to come by ...
0
votes
1answer
32 views

Transformation theorem, Cauchy distribution

I have derived the density for the ratio of two independent random variables,via the transformation formula. In this way: $V = X/Y $ and $ U = X $ inversion yields: $Y = U/V$ och $X =U$ , the ...
2
votes
0answers
29 views

How to scale “probabilities” to a given mean?

I have a set of scores $x_i$, $i=1,\ldots,N$ (mimicking probabilities, $0\le x_i\le 1$) and I want to transform them so that the result has a given mean $m$, while remaining in the interval $[0;1]$. ...
0
votes
1answer
18 views

Calculating probability of a time-series probability crossing a threshold

(Please feel free to suggest a better title -- I'm still not sure what to call this in the first place.) I'm having trouble getting my head wrapped around a time-series stochastics problem I've run ...
0
votes
1answer
28 views

Using joint probability density function to find the conditional probability of an event [on hold]

Obtain $P(2<Y<3 | X =1 )$ where the joint pdf of $X$ and $Y$ is $f(x,y) = (6-x-y)/8$ where $0 < x < 2$ and $2 < y < 4$. I have no idea about this question. can someone solve it for ...
2
votes
3answers
43 views

Easier way to solve conditional probability question?

Two digits are chosen at random from a table of random numbers containing the digits 0,1,2,...,9. Find the probability that the second number is 2, given that the sum is of the two numbers is greater ...
0
votes
0answers
27 views

Can someone help me with the following math question/dilemma?

I have a pool of objects that are randomly selected from a global object database. The objects certain numeric attributes: The objects from the pool are fed to users in real time Users will either ...
0
votes
0answers
38 views

What did I do wrong when using Jacobian transformation

A device containing two key components fails when, and only when, both components fail. The lifetimes, $T_1$ and $T_2$, of these components are independent with common density function $f (t) = ...
3
votes
2answers
28 views

correspondence between balls in compartments and integer vectors

I'm doing a self-review of probability and working through Ross' Introduction to Probability. The question is (Ross, ch2 number 51): suppose $n$ balls are randomly distributed into $N$ compartments. ...
1
vote
1answer
28 views

How to show that $\Phi(1-x)^{-1} =O(\sqrt{\log{x^{-1}}})$

In the middle of some proof, I have faced an expression $\Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}})$, where $\Phi(\cdot)^{-1}$ is a quantile function of the standard normal distribution and $x \in (0,1)$. ...
2
votes
1answer
22 views

Does “Expected Absolute Deviation” or “Expected Absolute Deviation Range” exist in stats and have another name?

So everyone is familiar with Variance and Standard Deviation from high school, but it seems no one has any familiarity with a philosophical justification for such weird, seemingly arbitrary measures. ...
1
vote
1answer
15 views

Find the probability $P[ x(t) \le 1]$ where $x(t)$ is a filtered Poisson process (rect pulses)

I can't understand the following question: "The random process x(t) is defined as $$x(t) = \sum_{n=- \infty}^{+\infty} rect(\frac{t-\tau_{n}}{T}) \quad ,\quad t \ \epsilon \ (R)$$ where {$\tau_{n}$} ...
2
votes
3answers
56 views

Conditional expectation of the sum of two random variables

I've got some difficulties in calculating the conditional expectation of the sum of two RV. I am not sure if I correctly formalized the scenario I am looking at. So I am trying to describe it first: ...
2
votes
1answer
21 views

The probability of getting a certain image by random pixelation

Well, seeing that I'm terribly bad at math I don't know how to solve this, I'll try to explain, excuse me if I sound dumb. Just suppose that I've got a photo/image with 320x240 resolution and 24 bit ...
1
vote
1answer
50 views

Expected number of changes of serves in a game of raquetball

Suppose a game of racquetball is being played, with players A and B. Assume further that A starts the play first, that the winner of a point serves the next point, and that the match lasts until the ...
0
votes
1answer
44 views

Probability of two teams meet up in FIFA tounament

**Second round is single elimination round. **Tournament is from 16 teams elimination follower by quarter finals,semi-finals and a final. **The losing semi-finalist contest a third place match. In ...
-5
votes
0answers
35 views

what is the probability of getting two heads twice in 5 tosses of two coins? [on hold]

If I have two identical fair coins and toss them both at the same time, what is the probability that after five tosses, two tosses resulted in both coins landing heads up?
0
votes
2answers
31 views

probability of a flipped coin

A fair coin is flipped three times. Let $A$ be the event that a head occurs in the first flip and $B$ be the event that exactly one head occurs. a) Find $p(A/B)$ b) Are $A$ and $B$ independent? ...
2
votes
1answer
43 views

Expectation related to Normal distribution and its density

Given $\sigma^2>0$. Let $Z\sim N(0,1)$ and $\Phi$ be the cumulative standard normal with density function $\phi$. I wish to show that $$ E\left(\frac{Z^2}{[\phi(\sigma Z)]^2}\Phi(\sigma ...
-1
votes
2answers
30 views

If pages in a book have an iid Poisson number of errors, in 10 pages what is the probability that exactly 3 pages have exactly 1 error?

Suppose the number of spelling error on any given page in particular book can be modeled by a Poisson distribution with $\lambda=2$, and assume that the number of errors on different pages is ...
-1
votes
0answers
39 views

Choosing random marbles until one is divisible by $X$ [on hold]

A box contains twelve marbles on which they are numbered by $1,2,3,...,12$. Now let $X$ represent the number of marbles you must choose with replacement until you obtain one with a number that is ...
1
vote
1answer
46 views

Probability of getting a right answer?

A student is taking a $4$ question multiple choice quiz with each question having $5$ options. What is the probability that he will get at least one question correct? P.S. Please keep answers at ...
0
votes
2answers
45 views

Expectation of CDF of continuous random variable $X$, evaluated at $X$

Given the continuous random variable $X$ with cumulative distribution function $F_{X}$, find $E[F_{X}(X)]$. Attempt at solution: I understand that the expected value, $E[X]$, of a random ...
0
votes
2answers
26 views

Convergence of running maximum of uniform random variables [on hold]

Let $X_1, X_2, ... X_n$ be an IID sequence of IID random variables that have a uniform distribution $(0,1)$. Let Max$(n) =$ max$(X_k:1\le k \le n)$, where $n\in \mathbb N$. How do I show that ...
1
vote
2answers
37 views

Confusion regarding the fixed point $p(x) =x$

Consider an empty urn. Now at each time, we are adding one ball to it, Either red or black, the probability of a red ball being added depends on $x$ ($x$ denotes the current fraction of red balls in ...
0
votes
2answers
50 views

Expected value and variance of max{x, y}

I've run into this problem while playing a game called Europa Universalis 4. I've done similar maths before in my studies so I'm pretty sure this should have an easy answer but I can't for the life of ...
1
vote
1answer
50 views

Calculate single “battle” outcome odds for RISK

I am trying to reproduce the values in this odds ratio table from Wikipedia. For all those unfamiliar with RISK, this is a game where units fight against each other via the roll of the dice: The ...
1
vote
0answers
24 views

change some element of a correlation matrix

I am working on correlation matrices. These matrices have the main property to be symetric , positive-semidefinite, have 1 on the diagonal and each of their elements is between -1 and 1. Let's say I ...
0
votes
0answers
16 views

Stopped strong Markov process again strong Markov?

Following setting: I have a right-continuous strong Markov process X in a right-continuous filtration >$\mathbb{F}=(F_t)$ and a P-a.s. finite stopping time $\tau$. My question is: Is the ...
1
vote
1answer
15 views

Convex and Concave Functions using Known Function Values

I am reading the classic Prospect Theory: An Analysis of Decision Under Risk (1979, Econometrica) by Kahneman and Tversky. I am not clear on something on page 278: ...
2
votes
2answers
41 views

Estimate the number of ants in a colony

A friend of mine gave me this weird problem I cannot solve. To estimate the number of ants in a colony an entomologist draws 5500 ants randomly from the colony, labels them with a radioactive isotope ...
30
votes
3answers
5k views

Making 400k random choices from 400k samples seems to always end up with 63% distinct choices, why?

I have a very simple simulation program, the sequence is: Create an array of 400k elements Use a PRNG to pick an index, and mark the element (repeat 400k times) Count number of marked elements. An ...
3
votes
3answers
58 views

Probability of a year which is not a leap year

If a 4 digit year is choosen randomly, what is the probability that it is not a leap year ? This problem has come in my exam and i have written like this I know that the number of four digit year ...
1
vote
2answers
37 views

what is the distribution of a uniform r.v. divided by an exponential r.v.?

Say $U=\frac{X}{Y}$. X and Y are independent with each other. X is a Uniform distribution r.v. $X\sim \mathcal{U}(0,1)$. Y is an exponential distribution r.v., $Y\sim\mathcal{Exp}(\lambda)$, whose pdf ...