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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-1
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1answer
15 views

Bayes theorem and conditional probability

I have a problem like this: Seventy-eight percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, ...
1
vote
2answers
13 views

Evenly filling spaces for a specific average value

Imagine I have $N$ spaces. Each space can be empty, or occupied. Given a fixed point value $x$ between zero and one, I would like to evenly populate the $N$ spaces such that $\frac{N_{\text{occupied}...
0
votes
1answer
36 views

Two length 3 straights vs. one length 5 straight. Which is more likely and by how much?

Using a well shuffled standard $52$ card deck, $2$ players (call them A and B) decide to play a game. They draw community (shared) cards (without replacement) until a winner for that hand is ...
23
votes
2answers
455 views

How long does it take a person with this “cheating” data-gathering strategy to achieve a desired result?

I have a perfectly fair coin, and my goal is to prove that it is unfair with a confidence level of 95%. In order to accomplish this, I will cheat. Whenever I fail to have enough evidence, I will ...
0
votes
1answer
17 views

Query on Uniform random variable MAX & MIN

Llet $U$ and $V$ be independent, continuous uniform random variables on the interval $\left[1,5\right]$. Find $$\Pr\left(\min\left(U,V\right)<2 \mid \max\left(U,V\right)>2\right)$$
1
vote
0answers
19 views

Is there a way to maximize this probability by taking the derivative of the cumulative normal distribution function?

I'm self-studying Brownian motion and encountered the following problem. I understand the author's solution, and it is clear why maximizing the right-hand side of the inequality provides such $t$ ...
0
votes
1answer
386 views

Probability Question - Pls help

A salesperson visits $k$ clients each day. The salesperson makes one and exactly one sale each day. The probability that the salesperson makes a sale to the $j$th client on a given day is $p_j$, ...
0
votes
0answers
8 views

Comparing log functions of CDFs and PDFs (related to order statistics) with non-log functions of the same

Let $f$ and $F$ denote the respective pdf and cdf of a probability distribution on $\mathbb{R}$. Take any natural $n\geq3$ and any real $a$ and $c$ such that $a\leq c$, and $\rho\geq0$. We want to ...
0
votes
3answers
46 views

Odds of 10 thrown dice landing all the same

What are the odds of throwing $10$ six sided dice and landing all the same number. Also, how many throws would I need to do to achieve a $100$ percent success of this happening. Is this even possible ...
0
votes
0answers
14 views

Predict the daily usage of Bandwidth of a Network

Context: I want to predict the daily usage of bandwidth of a network (consists a number of users) based on previous use . For example, I want to predict the amount of bandwidth during 8 pm to 9pm ...
1
vote
1answer
28 views

How to find a mean probability

A speaks truth in 75 cases just of hundred while B speaks truth in 80 cases out of hundred.Find the number of cases where they are likely to contradict. I did try working it out.So out of 200 ...
0
votes
0answers
19 views

Detailed explanation needed for basic query regarding expectation

I need to find the expectation of following random variable $$g=[\log_2(\frac{1+x}{1+y})]^+$$ where $[x]^+=max(x,0)$ and both $x,y$ variables depend on variable $z$. I know the conditional pdf's and ...
0
votes
1answer
31 views

How can I generate a sample from the distribution $P(x) = \frac{exp(-(x^2-\mu)^2)}{\sum_{\bar{x} \in \mathbb{R}}exp(-(\bar{x}^2-\mu)^2)}$

I wish to generate samples from generate a sample from the distribution $$P(x) = \frac{\exp(-(x^2-\mu)^2)}{\sum_{\bar{x} \in \mathbb{R}}\exp(-(\bar{x}^2-\mu)^2)}$$. The unnormalized probability is $\...
3
votes
2answers
41 views

How do I prove that for a random variable $X$, we have $P(X \le a) \le p$?

Specifically, suppose that $X$ is a random variable with properties $\mathrm{Var}(X) = 9$, $\mu = \mathbb{E}(X) = 2$, and $\max(X) \le 10$, (or $P(X \ge 10) = 0$). How can I prove the following? $$P(...
0
votes
1answer
16 views

maximum possible probability

75% of the customers of ACME Mutual Insurance have auto insurance, and 40% have homeowners insurance. What is the maximum possible probability that a randomly selected customer with auto insurance ...
0
votes
0answers
36 views

How can I prove that for a random variable $X$, we have $P(X \le \mu) = P(X \ge \mu)$?

Specifically, suppose that $X$ is a random variable with properties $\mathrm{Var}(X) = 9$ and $\mu = \mathbb{E}(X) = 2$. How can I prove the following? $$P(X \ge \mu) = P(X \le \mu)$$ It is also ...
0
votes
1answer
11 views

how to get a distribution using the moment generating function

we have that X has a normal distribution with mean μ and variance 4. and we have to get the distribution of $(x-μ)^2/4$. I tried this: Y=$(x-μ)^2/4$, then $M_{y}(t)=M_{(x-μ)^2/4}=e^{μ^2t}M_{(x^2-2xμ-...
1
vote
1answer
20 views

Conditional expectation and variance of exponential distributions

Okay, so here's two problems from my book; Problem 1) Let $f(x,y) = 2e^{-(x+2y)}$ $x,y>0$ Calculate $V[Y|X>3 \cap Y>3]$ Solution since the joint density can be factored out into terms ...
2
votes
1answer
55 views

probability of rank of a number

Suppose I have 10 sample means. I want to find the probability of rank of the population means using sample means. Therefore, I want to perform two experiments. First experiment: I pick one of the ...
1
vote
1answer
12 views

Chi-Squared Distribution

Let $Z_1, Z_2, Z_3$ be independent standard Normal R.V.'s. Which of the following has a Chi-Square distribution with 1 degree of freedom. $$ \begin{align} A) & & & \frac{Z_1^2, Z_2^2}{2} ...
0
votes
1answer
386 views

Conditional Probability and life expectancy

In a population of 100,000 females, 89.835% can expect to live to age 60, while 57.062% can expect to live to age 80. Given that a woman is 60, what is the probability that she lives to age 80? Using ...
2
votes
1answer
30 views

Is it true that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$?

I was wondering if we can show that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$ in general? Here $X_1$ and $X_2$ are independent but may not follow the same distribution. Any hint is much ...
2
votes
0answers
33 views

What does it mean if $cov(f(x1), f(x2))$ is positive in the context of LHS sampling?

If cov(f(x1),f(x2)) is positive, does that mean f is close to symmetric along x1 and x2? I am struggling to put this into understandable terms. Edit: The context is equation 6 in this paper: http://...
1
vote
2answers
21 views

Calculate probabilies based on given probability distribution

A mail-order company business has six telephone lines. Let $X$ denote the number of lines in use at a specified time. Suppose the pmf of $X$ is as given in the accompanying table \begin{array}{r|...
5
votes
1answer
91 views
+50

Example of a set and monotone class where monotone class is not a $\sigma$-algebra?

What is an example of a set $X$ and a monotone class $\mathcal{M}$ consisting of subsets of $X$ such that $\emptyset \in \mathcal{M}$, $X \in \mathcal{M}$, but $\mathcal{M}$ is not a $\sigma$-algebra?
0
votes
1answer
496 views

Figuring out probability of two random events both happening

So here's the problem: The table below shows the distribution of education level attained by US residents based on data collected during the 2010 American Community Survey: ...
1
vote
0answers
88 views

Exact Probability of reducibility of Bivariate Polynomials

I am considering polynomials of the form $$P(x,y)= \sum_{k=0}^n\sum_{l=0}^n a_{k,l}x^{k}y^{l}$$ where $n \in \mathbb{N}$. The coefficients $a_{k,l}$ are considered to be randomly generated from the ...
1
vote
0answers
23 views

How to Calculate the “Drop Off” of a Set

So I have never taken a formal class of statistics and this is likely just a case of me not knowing the right name for what I am looking for. Nonetheless, say I have a set of numbers in descending ...
0
votes
1answer
718 views

How to calculate t-value, given degrees of freedom and $\alpha$.

While solving problems, we can look up physical t-tables or use a statistical analysis software like R to calculate t-values. But how do we actually calculate these values ? What is the algorithm ...
-1
votes
2answers
56 views

probability of getting an erdos number once published [closed]

Can I know that I don't have an erdos number once I published, what the probability is of getting an erdos number with "random" coauthors or can I formulate the probability of having a finite erdos ...
0
votes
0answers
23 views

distribution and density of maximum minus element

I am a bit rusty in probability, and for a project I am studying the random variable $Z = \max(X_1, \ldots, X_n) - X_i, i = 1, \ldots, n$ where the $X_i$ are positive independent random variables. In ...
1
vote
0answers
39 views

The Spacing of $e$ and $\pi$ Segments Within the Decimal Expansion of $\pi$

I discovered something seemingly very improbable today when I was searching for segments of $e$ and $\pi$ within the decimal expansion of $\pi$. I searched for $314159265$ and found it starts at the ...
2
votes
3answers
227 views

What are the odds of flipping a coin 100 times and seeing HHHHT? [on hold]

What are the odds of flipping a coin 100 times and seeing exactly four consecutive heads? Any more than four heads in a row, such as "HHHHH" would not be considered a string of four consecutive heads. ...
-3
votes
1answer
27 views

Conditional probability using set notation [on hold]

Got this wrong on a quiz and i don't have the answers. Need to figure this out for a test coming up. \begin{align} P(A) &= 0.75 \\ P(B\mid A) &= 0.9 \\ P(B\mid A^c) &= 0.8 \\ P(C\mid A\...
2
votes
0answers
62 views

When to stop pumping up balloons?

Yesterday I acted as a volunteer in a psychology/neurology experiment where one of the trials consisted of playing a computer game in which you had to click the mouse to pump up a balloon. For each ...
-2
votes
1answer
19 views

Suppose X and Y have joint density f (x, y) = 2 for 0 < y < x < 1. Find P (X − Y > z). [on hold]

Suppose X and Y have joint density f (x, y) = 2 for 0 < y < x < 1. Find P (X − Y > z). Solution is (1-z)^2
1
vote
1answer
26 views

Probability Sum

A purchasing agent must decide to accept or reject an incoming shipment of machine parts. The agent wishes to do either of the following: a1: Accept the shipment a2: Reject the shipment The fraction ...
0
votes
2answers
34 views

Find the limit of the probability of uniform random variable?

Let $X_1 ,X_2 ,X_3 ,…$ be a sequence of i.i.d. uniform $(0,1)$ random variables. Then, calculate the value of $$\lim_{n\to \infty}P(-\ln(1-X_1)-\ln(1-X_2)-\cdots-\ln(1-X_n)\geq n)?$$ My work: Since ...
-1
votes
1answer
53 views

A binomial sum identity

Let \begin{align*} f(n, r, \pi, k) &= \sum_{z=0}^{n}\sum_{s=0}^{r}\binom{z}{s}\binom{n}{z}\binom{n-z}{r-s}(-1)^{r+s}\left(\frac{\pi}{1-\pi}\right)^{r/2-s}\pi^{z}(1-\pi)^{n-z}z^k \end{align*} I am ...
2
votes
0answers
36 views
+50

Expected score from threshold with number deletions

We play a game where a sequence of $n$ numbers is drawn uniformly from $[0,1]$, and we need to set a threshold $0\leq a\leq 1$. For every number that is at least our threshold, we get $a$ points but ...
0
votes
2answers
36 views

Random probability

So the story is my friend was playing Runescape and he was trying to get an item drop that had a ${1\over 128}$ drop rate so on average every $128$ monsters he slays one of the items will drop.he ...
-1
votes
0answers
16 views

recursive definition for two mutually exclusive events [on hold]

How do we write recursive definitions for two mutually exclusive events ? Can anyone explain with some examples as how do we come up with solutions in case of exclusive events ? SO finally i add ...
1
vote
0answers
16 views

Probabilistic Method/Model for Traffic Flow

Context: Given a network system or a traffic system with some value related to the system. Question: Which probabilistic methods, model, distributions are used frequently to predict a event (for ...
0
votes
1answer
387 views

Sub sigma algebra and probability spaces — definition

I am reading this book and I am a bit lost with the definitions because they are not provided and I can't seem to find it online: Let $L_2(\Omega,A,P)$ be a probability space such that $f \in L_2$ ...
-4
votes
0answers
39 views

Length of left stick [on hold]

Break the stick into 3 pieces and what would be the expected length of left stick? I need answer for this to verify my answer. Can somebody give me the answer? My thoughts: My answer is 1/4. First ...
2
votes
1answer
37 views

determine the distribution of the random variable $Y=\Sigma_{k=1}^{\infty}kX_k$

Fix $p \in (0,1)$ and consider independent Poisson random variables $X_k$, $k \geq 1$ with $\mathbb E[X_k]=\frac{p^k}{k}$. Verify that the sum $\Sigma_{k=1}^{\infty}kX_k$ converges with probability ...
2
votes
1answer
70 views

Creating unusual probabilities with a single dice, using the minimal number of expected rolls

Problem I want to create an 'event' with probability of $\frac{1}{7}$ with a single dice as efficiently as possible (to roll the dice as little as possible). To give you some better understanding of ...
0
votes
0answers
20 views

Variance: logical/mathematical meaning [duplicate]

$$\operatorname {Var} (X)=\operatorname {E} \left[(X-\mu )^{2}\right]$$ Is the formula of variance. But if you think of it, you can assume that square was introduced just that something other than ...
43
votes
8answers
10k views

Lebesgue integral basics

I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ...
14
votes
10answers
9k views

Proving Pascal's Rule : ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$

I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$. I suppose I could use the counting rules in probability, perhaps combination= ${{n}...