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
votes
0answers
16 views

How to prove or disprove the following?

There are two questions I want to ask for: If $X_n \rightarrow_{a.s.} X$ and $Y_n \rightarrow_{a.s.} Y$, then $X_nY_n \rightarrow_{a.s.} XY$ If $X_n \rightarrow_{L^r} X$ and $Y_n \rightarrow_{L^r} ...
4
votes
5answers
150 views

$A+B+C=2149$, Find $A$

In the following form of odd numbers If the numbers taken from the form where $A+B+C=2149$ Find $A$ any help will be appreciate it, thanks.
2
votes
2answers
31 views

Probability of Boys and Girls in Row

Ten male friends and six female friends line up next to the bus stop in a row. Everyone just positions themselves at random. What is the probability that no two females are sitting next to each other? ...
1
vote
2answers
32 views

Maximum value of the product of probabilities

I came across a confusing probability problem. It reads as follows: Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$. If $P(\cdot)$ denotes the ...
0
votes
0answers
9 views

Multivariate gaussian and average covariance matrix

Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in ...
3
votes
1answer
62 views

How long before the prey can escape?

I've (sort of) come across the following problem in my research. The actual scenario is a little abstract to explain, so I'm rephrasing the problem in terms of a predator/prey scenario. I'm tagging ...
1
vote
1answer
506 views

How to prove convergence in mean implies uniform integrability?

My class notes and wikipedia both say that $X_n \xrightarrow{L^1} X$ $\Leftrightarrow \; X_n \xrightarrow{P} X$ and $X_n$ are uniformly integrable. I am trying to work through the proof. I am able ...
0
votes
1answer
26 views

Probability of atleast 2 people out of 3 having same birthdays

I want to confirm that the answer will be ( 365 × Combination(3,2) × (1/365)^2 ) + ( 365 × Combination(3,3) × (1/365)^3 ) Please refer to this answer for reasoning Probability of 2 people having ...
0
votes
1answer
17 views

Finding expected number of trials until we get head given density function?

Suppose we flip a coin with a random probability of Heads $P$ that has density $f(p) = 6p(1−p),\; p \in [0, 1]$. If we keep on flipping this coin until we get a single Heads, what is the expected ...
-1
votes
1answer
19 views

Question about finding expected value and variance of x given the mean.

Suppose Y is distributed as an exponential random variable with mean 0.5 and given Y = y, X is distributed as an exponential random variable with mean y. What is E(X) and Var(X)?
-3
votes
1answer
37 views

Probability of 2 people having same birthdays [on hold]

Was wondering if answer will be: number of days in year × probability of same birthday $$P = 365 \times \left(\frac{1}{365}\right)^2$$ I promise there is a part 2 of this question, hence please ...
1
vote
1answer
29 views

Question about package of cookies that is a random variable

The weight in grams of package of cookies is a random variable with expected value of $300$ grams $\color{blue}{A)}$ assume that X is normally distributed with standard deviation of $15 $ grams ...
0
votes
2answers
36 views

Get the distribution of $X|Y=y$ given this joint probability density function

Given the joint probability density function $f(x,y) = \lambda^2 \exp(-\lambda y)$ with $0 < x < y.$ How do I get the distribution of $X|Y=y$ ? Thanks in advance!
0
votes
0answers
27 views

Probability of a run of *n* or more of some color from a subset of colors drawing without replacement?

I recently asked the question "Probability of a run of k or more of a subset of categories in m multinoulli trials?" with a very nice answer from member Tad. I'm trying to extend a result from a ...
0
votes
1answer
13 views

Probability of getting a bingo pattern with a certain number of balls drawn?

Let's say you have a fixed pattern on a standard 5x5 bingo card using all the standard bingo rules (center is a free space, B column has numbers 1-15, I has 16-30, etc. to 75). How can you calculate ...
-1
votes
2answers
31 views

Probability of choosing same number

There are four friends – Adam, Bella, Christopher and Drew. All of them are asked to choose any number in their mind. Now what is the probability that every one of them has the same number in mind? ...
0
votes
1answer
19 views

Quick question concerning the sum of random number of random variables given mean and variance and average

$\DeclareMathOperator{\cov}{cov}$The problem is: Let $X_1, \ldots, X_n$ be independent random variables with mean $µ$ and variance $σ^2$. Let $X¯$ be the average of these n random variables. Find the ...
1
vote
3answers
6k views

Chance of getting six in three dice

I am having a hard time wrapping my head around this and am sure that my answers are wrong. There are three dice. A. Chance of getting exactly one six on the three dice. (1/6) * 3 = 1/3 B. Chance ...
2
votes
1answer
128 views
+50

Find a probability density

I am going through a paper trying to understand all the single steps, but I got stuck. I need to calculate $$p(x+\delta t) \mid x(t), t)= \int p(x(t+\delta t) \mid \mu , x(t), t)p(\mu\mid x(t), t) ...
1
vote
0answers
43 views

How to find the density of $Y=g(X)$ in this case?

I have a vector $X=(1,X_2,X_3)$, where $(X_2,X_3)$ is a random vector in $\mathbb{R}^2$. Now consider $Y=g(X)=X/\|X\|$. What is a density function of $Y$ with respect to the uniform spherical ...
3
votes
0answers
31 views

Convolution of probabilities

It is a well known fact that for a random variable $Z=Y_1+Y_2+...+Y_n$ where $Y_i$ are independently distributed then the probability density function of $Z$ is the convolution of the density ...
1
vote
1answer
34 views

Definition of n independent event and example

Given a finite set of events $A_1,\dots,A_n$, the events are said to be independent if and only if for any subset of indices $I$ we have: $$\mathrm{P}\left(\bigcap_{i\in I} A_i\right)=\prod_{i\in I} ...
0
votes
1answer
22 views

Probability of an event if the sample space has identical elements

Suppose we have a box, with only one small hole. Suppose 10 distinct black balls and 20 distinct white balls are put in the box. Now, in a random draw of 1 ball, the probability that the ball drawn is ...
2
votes
3answers
59 views

Can you get a fair coin flip by rolling a fair, 5-sided die a finite number of times?

Can you get a fair coin flip by rolling a fair, 5-sided die a finite number of times?
0
votes
1answer
47 views

Probability as a function of time

I was really wondering when I have to select any one out of the n options available - the probability of selecting A (let's say) is 1/n. But then I'm confused. When I (or anyone/anything else) bring ...
-1
votes
1answer
22 views

Risk Reduction equation

If, after a point in time, your risk of an event falls by 50% in 1 year and then by 100% in 15 years, can someone help me with the equation that will look at your risk reduction to date for any given ...
1
vote
2answers
33 views

Reverse Bernoulli Trial?

I'm struggling to figure out how to do what I think would be called a reverse Bernoulli trial, essentially: How many coin flips must I make to have a 75% change of getting three heads? First of ...
0
votes
1answer
19 views

Probability set function of the random variable $X$

Let a point be selected from the sample space $S = (0,10)$. Let $C \subset S$ and let the probability set function be: $$P(C) = \int_C \frac1{10}\ \mathsf dx$$ Define the random variable $X$ by: ...
0
votes
1answer
30 views

Probability of 4-number matching in a lottery in two different situations?

In some lottery, 7 numbers are drawn and each of them from numbers ${\{1, \dots, 45}\}$. To win "Division 6" means to have 4 of 7 drawn numbers. The order of drawn numbers doesn't matter. My ...
0
votes
0answers
8 views

How to illustrate that the first-fit algorithm for bin packing problem uses STRICTLY MORE bins when only one object becomes larger?

In the bin packing problem, objects of different volumes(lie in $[0, 1]$) must be packed into a number of bins(each of capacity 1). The first-fit algorithm attempts to place the item in the first bin ...
0
votes
0answers
13 views

Conditional density of degenerate multivariate normal

Let $X_{12},X_{13},X_{14},X_{23},X_{24},X_{34}$ be identically normal $N(\mu,\sigma^2)$ such that every linear combination among $X_{ij}$'s is also normal, $corr(X_{ij},X_{rs})=\rho$ if ...
1
vote
1answer
39 views

Are infinite-dimensional singletons measurable?

Consider the wiener measure space $C[a,b]$ of all real-valued continuous functions on $[a,b]$ with the wiener measure $\mu$ on the $\sigma$-algebra $\mathcal{A}$ of Carathéodory measurable sets in ...
-1
votes
1answer
51 views

A conditional probability question [on hold]

Let A and B two events and if $P(A)=0.5$ and $P(B)=0.4$ what is the $P(B\mid A^C)$?
3
votes
1answer
25 views

Cancellation law of equal in distribution

I came across this gem while discussing with my friends, If $X$ and $Y$ are two real valued random variables (not necessarily independent) that satisfy $$X =^d X+Y$$ (where $=^d$ means equal in ...
2
votes
1answer
43 views

Basic probability and counting methods

A somewhat geeky problem has been on my mind the last few days: In my accommodation at Uppsala, there are 12 rooms to a floor. I discovered the other day that another British girl whom I know lives ...
0
votes
2answers
25 views

Understanding different definitions of bayes theorem

I am taking course on probability and reading about bayes theorem. In Sheldon Ross' book, it given as $$P(E) = P(E|F)P(F) + P(E|F^C)P(F^C)$$ with note: Equation above states that the probability of ...
4
votes
2answers
73 views

“Mastermind”-esque safe opening problem.

I read this interview question for a trading job and it seems quite difficult. What is the technique to solving it? You have a safe with six digits and a light. You can input a code, if you have ...
2
votes
2answers
28 views

Statistics question on basil bush random variable

The height, $H$, in meters of a basil bush is a random variable with the probability density function $f_{_H}(t)=e^t,\;0\leq t\leq H_0$ such that $H_0$ is the maximal height. $\color{blue}{(1)}$ I ...
0
votes
1answer
15 views

Inner product estimator - random variable

I'm curently working on the functional space $L^2(\mathbb{R}^n,B(\mathbb{R}^n),\mathbb{P}_X)$ where $\mathbb{P}_X$ is a probability measure. If I generate randomly $N$ realizations of $x_i$ following ...
1
vote
3answers
184 views

The probability that in a word made from a set of 16 letters exactly two are repeated

A word of 6 letters is formed from a set of 16 different letters of English alphabets (with replacement). Find the probability that exactly two letters are repeated. The answer is ...
2
votes
1answer
26 views

Coding Theory - Probability that word received has distance of at most 1?

Suppose the codeword x = 101101 is transmitted over the binary symmetric channel, with symbol error probability p. What is the probability that the word received has distance at most 1 from x? ...
1
vote
0answers
27 views

On the Chernoff bound

Recently, I saw the Chernoff bound written as follows. Let $X_1,X_2,\ldots,X_n$ be drawn i.i.d. on alphabet $\mathcal{X}$ and let $f:\mathcal{X}\to [0,1]$ be any function. Let $\mathbb{E}f(X_1) = ...
0
votes
0answers
20 views

Probability of word appearing in a new document given the probability it appears in earlier documents [on hold]

Suppose I have a set of keywords and 10 documents . I can count the frequency and probability of a given keyword occuring in each of these documents . What will be the probability of the same keyword ...
1
vote
4answers
52 views

Probability on selecting balls

If I have B black balls and W white balls in a bag, what is the probability that the last one I select is white? How shall I solve this problem? I am not sure how to make a start, is it correct ...
3
votes
1answer
35 views

convergence in probability: speed of convergence

I am not sure if the title appropriately describes the question. I will appreaciate any ideas. Suppose $\{X_n:n\geq 1\}$ is a sequence of random variables defined on a common probability space. ...
0
votes
1answer
16 views

Hitting times of a biased continuous time random walk

Let $X_{s \geq 0}$ be a continuous time random walk on $\mathbb{Z}$, i.e. waiting times between jumps are exponentially distributed with mean one. The random walk is biased: $\mathbb{P}(X_s\text{ ...
0
votes
1answer
412 views

Greatest of three random variables

Assume that we have $3$ not equal random variables $(A, B, C)$. If we know that $$Pr(A>B)=x, \quad Pr(A>C)=y, \quad Pr(B>C)=z$$ What is $Pr(A$ is the greatest one)? I know that $Pr(A$ is ...
0
votes
1answer
21 views

Sums of partially dependent Bernoulli random variables

I am looking for any kind of Chernoff type large deviation bound for the following random variable: $$X = \sum_{i=1}^NX_i$$ where each $X_i$ is an identically distributed Bernoulli random variable ...
5
votes
6answers
5k views

If I buy 2 lottery tickets do I double my chance of winning?

There's a lottery. There are 6 balls chosen randomly from 49 and you have to match all the balls to win. I buy one ticket. If I buy two tickets with different numbers for the same draw, do I ...
2
votes
0answers
73 views

Of strings and substrings: A problem of probability

Problem Let $\Sigma=\{a, b\}$. Let $\Sigma^*$ denote the Kleene star of $\Sigma$: \begin{equation*} \Sigma^* = \{\varepsilon, a, b, aa, ab, ba, bb, aaa, aab, \ldots\} \end{equation*} where ...