Questions tagged [probability]

For basic questions about probability and the questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities. For questions about the theoretical footing of probability (especially using [tag:measure-theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

Filter by
Sorted by
Tagged with
0
votes
0answers
7 views

Very quick question about random variables (Formal writing).

I've came across this confusion while reading this question: The time until Sam receives his first email is exponentially distributed with parameter $4$. The time between the first email and the ...
0
votes
1answer
27 views

Law of total probability proof

I am trying to prove $P(A_1\cap A_2\,\cap ...\cap A_n)=P(A_1)P(A_2|A_1)P(A_3|A_1\cap A_2)\,\cap ... \cap\,P(A_n|A_1\cap A_2\,\cap ...\cap A_{n-1})$ assuming that the conditional probabilities exist. ...
-2
votes
0answers
22 views

K consecutive heads in N tosses USING COMBINATRONICS

I would like to understand how to approach questions as: Find the Probability of observing K consecutive heads in N tosses, but I am interested only in the combinatorics/permutations approach, since I ...
1
vote
1answer
44 views

Is $(y,T)=(7,8)$ the only solution in positive integers to $\frac{F(y)}{F(T)}=\frac12$, where $F(x)=x(x-1)(x-2)(x-3)$?

Define $F(x)=x\cdot(x-1)\cdot(x-2)\cdot(x-3)$. Let $y,T$ be positive integers. Is it then true that $y=7$, $T=8$ is the only solution to this equation? $$\frac{F(y)}{F(T)}=\frac{1}{2}$$ Motivation ...
2
votes
0answers
12 views

Two sequences of measures $(\mu_n)_{n \geq 0}$ and $(\nu_n)_{n \geq 0}$ weakly converging to singular measures $\mu$ and $\nu$

I have two sequences of positive finite measures $(\mu_n)_{n \geq 0}$ and $(\nu_n)_{n \geq 0}$ that are absolutely continuous w.r.t. Lebesgue measure, more precisely \begin{equation*} \mu_n(dx) = a_n^...
1
vote
0answers
33 views

5 consecutive heads in 25 coin tosses with linearity of expectation

I am reasoning around the linearity of expectation. If for example, I want to know the expected number of a pair HH in 25 tosses (note that HHH has 2 possible HH pairs) I could use linearity of ...
0
votes
2answers
36 views

Finding Expectation and Variance of a Subset

You have a box containing $20$ LEGO bricks, $4$ of which are broken. You randomly take $3$ bricks out of the box. Let $X$ be the number of broken LEGO out of the $3$ bricks. Find $\mathbb E[X],\ Var (...
1
vote
1answer
61 views

If P(A|B)=1, then is it correct that P(A|B,C)=1 under the assumption that P(B,C)>0?

If $P(A\mid B)=1$, then what can we say about $P(A\mid B\cap C)$? Is it 1 or not? The condition says that if B occurs, then A occurs a.s. Then, if B and C occur, this implies that B has occurred and ...
0
votes
4answers
26 views

Find probability of two event given probability of intersection of complement with other event

Given $P(E_1' \cap E_2)=a$, $P(E_2' \cap E_1)=b$, find $P(E_1)$ in terms of $a$ and $b$ CUCET 2021 Came in a test I gave today, here is what I did: $$|P(E_1' \cap E_2) - P(E_2' \cap E_1)| =| P(E_1) -...
0
votes
0answers
39 views

Probability of at least $x$ successive wins in $n$ games

I play $n$ games of chess against my friend. I have a probability $p$ of winning, my friend a probability of $1-p$ of winning a single game. Assume games are independent. What is the probability that ...
0
votes
1answer
21 views

Bivariate Normal Distribution and sub vector

Let $$(X,Y)\sim N(\mu^\rightarrow ,\Sigma)$$ we cannot assume anything about the dependancy between X and Y. Can we assume the following? $$X\sim N(\mu_x,\sigma ^2_x)~,~Y\sim N(\mu_y, \sigma^2_y)$$...
2
votes
1answer
55 views

2 friends shooting

You and a friend play a shooting game. The chances of you hitting 1st and 2nd shot are 0.3 and 0.5 respectively. The chances for your friend are 0.4 and 0.4 respectively. Who has a better probability ...
3
votes
3answers
108 views

If 12 distinct balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls.

This is a bonus question asked in a competitive examination. It all choice are wrong. I will put the question, then the solution I got from the internet and then my solution. Please help to figure out ...
-1
votes
0answers
16 views

What is the best way to calculate an accurate representation of rarity? [closed]

I am working on a site that also ranks NFt's and I think this question is very interesting to get into the best way of calculating rarity and chance of things happening. My progress is getting the ...
0
votes
1answer
32 views

Defect in Engineering Product

I have an assignment Problem but I am unable to determine the answer to the following question Consider a rare defect with a certain type engineered wood product. During production there is a small ...
0
votes
0answers
9 views

Maximal cardinality of a set of pairwise independent variables all dependent with another variable

I suppose that I have a multinomial random variables $V$ with $n$ states. I would like to know how many boolean variables $B_i$ can possibly exist being each dependent with $V$ while being two by two ...
-2
votes
1answer
24 views

Please help me solve this exponential distribution problem using the mean [closed]

The time it takes for an economy to recover from a natural disaster is exponentially distributed with a median of 4 years. a) What is the parameter for this distribution function (a symbol and value ...
1
vote
0answers
64 views

Existence of product of uniformity variables below 1

Let $X_i$ be chosen uniformly and independently from $(0,1)$ for $i = 0, 1, 2, 3, \cdots$. For each such $i$. define $$Y_{i}=X_{0} \cdot X_{1}^{-1} \cdot X_{2} \cdot X_{3}^{-1} \cdot \ldots \cdot X_{i}...
2
votes
1answer
37 views

Ordering of independent events

Probability of Team A scoring $1$ goal is $P(A=1)=0.7,$ Probability of Team $B$ scoring $1$ goal is $P(B=1)=0.5.$ Events A and B are independent, and the teams do not score more than 1 goal each. On ...
3
votes
3answers
39 views

Probability of One Event Less Than Probability of Second Event

I am having a bit of trouble proving some cases when one probability is smaller than the other probability for all positive integers $a, b$, so some suggestions would be appreciated. Here is the ...
2
votes
1answer
22 views

Do formulas of Shannon information remain valid by a conditioning operation?

Do equality remain valid after the operation of adding conditioning events to a linear Shannon equation. If so, how do you prove this? Let $X,Y,Z$ be random variables. Let us tentatively call (...
0
votes
0answers
36 views

Covariance and Correlation by indicator method.

A question for the interview. An urn contains 90 marbles, of which there are 20 green, 20 black, and 50 red marbles. Tom draws 10 marbles without replacement, and Jerry draws 10 marbles without ...
1
vote
1answer
19 views

Evaluating a definite integral of the product of a power function and an arbitrary density function

I cannot seem to get any traction on evaluating an integral of this form: $$\int_{0}^1f(x)x^ydx,$$ where $f(x)$ is some arbitrary density function. Is there an analytic solution to this form? Are ...
2
votes
2answers
48 views

What's the probability of getting any one face at least $m$ times, when throwing a $k$-sided die $n$ times?

Say we have a $k$-sided die, and we want to throw it $n$ times. What's the expression for the probability of getting any one face up at least $m$ times, where $m \in \{0, ... , n\}$? Edit: The ...
1
vote
2answers
21 views

If $f(x,y) = cx $ and $0<x<y<1$ prove that $E[x] = 1/2$ where X,Y are positive random variables

I worked as such: $$\int_0^1\int_0^y cxdxdy=1 \rightarrow c=6$$. Also: $$f_x(x) = \int_x^1 6xdy = 6(1-x)x$$ So now I need: $$E[X] = \int_0^yx(6(1-x))dx$$. This is y-dependent , the result is not. What ...
0
votes
1answer
36 views

How are Fraction Constants Integrated in Definite Integrals

I have a question about double integrals (integration). For some reason, I find it much more confusing to integrate when I see fraction constants (6/5 from below). I know what to do with the x and y ...
5
votes
1answer
44 views

Composite of uniform distributions

If $X_1$ is uniform $(0,1)$ and $X_2$ is uniform $(0, X_1+1)$, what is $X_3$, which is characterized as uniform $(0, X_2+1)$? I simulated $X_3$ and got the following graph but found it hard to compute ...
0
votes
0answers
16 views

Use of Integrals as a Methodology of Solving Probability Problems

I was recently trying to solve for the probability of an event occurring when, due to lack of precise information, I found that there were two values in between which $P$ (probability of the event ...
0
votes
0answers
38 views

How many outcomes for 16 light switches

So my schools asking, what is the outcome for 16 switches, but they gave us counters, and that'll take a long time. So I'm trying to find a formula to calculate it easier. I saw on a different website ...
0
votes
0answers
21 views

order statistic of non-standard uniform distribution to beta distribution

If Xi∼U(0,1), then the jth ordered statistic is given by, X(j)∼Beta(j,n+ 1−j). I however do not understand how if Xi∼U(0,θ) then 1/θX(j) ∼ Beta(j,n+ 1−j). I have been looking for proofs online but i ...
0
votes
1answer
29 views

I need to calculate $P(A(50-T)\geq9)$, with $T\sim\exp(2)$ and $A\sim\exp(1)$ and $T$ and $A$ are independent.

So I need to calculate $P(A(50-T)\geq9)$, with $T\sim\exp(2)$ and $A\sim\exp(1)$ and $T$ and $A$ are independent. Also for integrating the lower limit can't be below zero. I think I found the joint ...
-3
votes
0answers
22 views

Normal distribution question of probability [closed]

The width of a slot of forgings is normally distributed with a mean of $0.9000$ and a standard deviation of $0.0030$. The specifications limits were given as $0.9000\pm 0.0050$. What percentage of ...
2
votes
1answer
53 views

Two piles of 26 cards

There is a 52-card deck. The deck is split into two piles of 26 cards each. I need to find the following probabilities: $1)$ Each pile contains two aces $2)$ The first pile contains more hearts than ...
3
votes
3answers
51 views

A class has divided 6 students randomly into teams A and B. What is the probability that 3 students from team A will come first, second and third?

My take on the problem is, considering the players to be indistinguishable individually other than by their teams i.e. the players are A,B,A,B,A,B. They can be arranged in $\frac{6!}{3!3!}$ ways. ...
1
vote
1answer
32 views

Loot Box Probability Question - Probability of Rare Items within Stated Constraints?

If there's a 9% chance of a rare item being revealed per loot box, and the consumer has 7 loot boxes, what's the probability of 0-8 rare items? I believe that binomial distributions are involved but ...
1
vote
1answer
44 views

A bag contains n white and 2 black cards.

A bag contains n white and 2 black cards. Balls are drawn one by one without replacement until a black is drawn. If 0,1,2,3,... white balls are drawn before the first black, a man is to receive $0^2$, ...
1
vote
0answers
29 views

When is the expectation of the maximum bounded

Let $\epsilon\equiv(\epsilon_1,\dots, \epsilon_J)$ be a random vector. Let $F$ be the probability distribution of $\epsilon$. Assumption 1: $F$ has full support on $\mathbb{R}^J$. Assumption 2: $F$ ...
2
votes
2answers
40 views

Expected Value and Exponential memoryless property

Given 8 runners, the time until they reach the finish line is distributed like so $X_1..X_8 \sim Exp(1)$ and they are independent from each other. Let $T_1$ denote the time of the runner who got to ...
0
votes
0answers
24 views

Urn A contains 5 cards numbered from 1 to 5 and urn B contains cards numbered from 1 to 4.

Urn A contains 5 cards numbered from 1 to 5 and urn B contains cards numbered from 1 to 4. One card is drawn from each of these urns. Find the probability function of the number which appears on the ...
0
votes
0answers
24 views

Corner case of independence of RVs

Suppose you have a random variable $X$. Intuitively one thinks that the random variables $X$ and $2X$ will not be independent. But I think it is not so if $X$ follows a point mass distribution, that ...
1
vote
1answer
65 views

Expectation of the product of random variables

An urn contains $n$ cards marked from $1$ to $n$. Two cards are drawn at a time. Find the mathematical expectation of the product of the numbers on the cards. I tried to answer this question and ...
0
votes
2answers
49 views

Probability that a 5-card poker hand is a straight

I'm trying to find the probability that a 5-card poker hand contains 5 numbers in a numerical sequence. For the first card, there are 52 options. For the second, there are 4 on either side of the ...
4
votes
1answer
58 views

Given $n>2$ and $1<k<n$. Is it possible to exist $n$ events(probability>0), s.t. for any $k$ among them are independent while any $k+1$ are not?

For example when $k=2$, consider $n+1$ disjoint events : $E=\{ e_0,e_1,...,e_n\}$ with the probability that $P(e_0)=a,P(e_i)=b,1\leq i\leq n.$ We define $n$ events $E_1,E_2,...,E_n$ such that $E_i=\{...
1
vote
0answers
39 views

Minimizing the variance in a variant of bagging Weighted Aggregation(Wagging)

In our machine learning course we have learned Bagging, wherein A variant of bagging call Weighted Aggregation is introduced, where the result is a weighted sum of all the estimators instead of ...
0
votes
0answers
23 views

prove S as a probability distribtuion summation to 1 on a given variable [closed]

s(A, B|C, D) as: q(A, B|C, D) = Sum(p(A|C', B, D).p(C'|D).p(B|C,D)), Sum on C' in the above equation. show s(A, B|C, D) as valid probability distribution. Please show pointers on how to proceed. Thank ...
0
votes
0answers
27 views

How do I calculate the probability of getting a flush from a 52 card deck using mathematica? [closed]

I know how to calculate this with pen and paper but would like to do the same in mathematica. However I'm new to mathematica so I have no idea where I'm supposed to start. Watched a couple of YouTube ...
0
votes
1answer
31 views

Alternative solution to license plates question

I'm studying probability and I tried to think an alternative approach to solve this question. The statement is as follows: How many different 7-place license plates are possible when 3 of the entries ...
0
votes
1answer
26 views

Joint likelihood of n samples iid from a binomial distribution vs joint probability, and the lack of a binomial coefficient

Let's assume I have 4 observations with each observation is modelled as a bernoulli trial with probability $p$. Sucesses are labelled as 1, failure is 0. My observations $(x_1, x_2, x_3, x_4)$ are as ...
1
vote
0answers
23 views

Closed-form formula for $u(t):= \mathbb E[h(X_1)h(tX_1+(1-t^2)^{1/2} X_2)]$, where $(X_1,X_2,\ldots,X_d)$ is uniform on sphere and $h(x):=(x+c)^k$

Fix $c \in \mathbb R$ and an integer $k \ge 0$, and consider the function $h:\mathbb R \to \mathbb R$ defined by $h(x) := (x+c)^k$. Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of ...
2
votes
1answer
40 views

Uniform probability measure on integers and arithmetic progressions

Does there exist a probability measure on the integers such that, the probability of any two arithmetic progressions with the same difference part, is the same? We assume the probability measure is ...

1
2 3 4 5
1826