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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6
votes
3answers
34 views

distribution of one random over the sum of random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables with $X_i\sim Gamma(\alpha_i,\beta)$. Define $U_i=\frac{X_i}{X_1+\cdots+X_n}$ for $i=1,2,\ldots,n$. Show that $U_i\sim ...
0
votes
0answers
28 views

The Gambler's Ruin without using random walks

This is more of a doubt. I understand that this problem can be described with Markov chains and the recursion solved without much trouble. However I've seen that some people casually say that $$ ...
2
votes
1answer
28 views

How to compute the expected value of one random variable over sum of iid random variable

If $X_1,\ldots,X_n$ are independent identically distributed positive random variables, prove that $E(\frac{X_i}{X_1+\cdots+X_n})=\frac{1}{n}$, $i=1,\ldots,n$. Can someone give me a hint?
-2
votes
1answer
24 views

Probable winner of last coin game of a series, where winner from one game has disadvantage the next game?

Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner is the first person to obtain a head. They play this game several times, with the stipulation that the loser ...
1
vote
2answers
37 views

Random walk in one dimension with different probabilities

As the title suggests, I'm concerned with a typical random walk problem, where the probability to go right is $p$ and the probability to go left is $q=1-p$. I was trying to find the probability of ...
1
vote
2answers
36 views

$X$ and $Y$ are independent Poisson$(\lambda)$, $\lambda\sim\mathrm{exp}(\theta)$. What is the conditional distribution for $X$ given that $X+Y=n$?

To clarify, the parameter $\lambda$ is a random variable with exponential distribution and parameter $\theta$. Can someone please tell me if I've correctly computed the conditional distribution for ...
-5
votes
1answer
27 views

Expected Value and Expected Variance Probability

Suppose a production line manufactures ball-bearings with a radius that is uniformly distributed between 1.8mm and 2.2mm. What is the probability of (a) the expected value of the volume, and (b) the ...
0
votes
1answer
23 views

probability distribution of the winning amount

Be A_n the event that a worker spends to process certain component with probabilities according to the table below: For each piece processed, the worker earns a fixed US 2.00, but if he processes ...
-1
votes
0answers
23 views

Probability, normal distribution, car collision

There is a question in the book Principles of Statistics by M.G. Bulmer which I'm stuck on. Here goes: The reaction time of two motorists A and B are such that their braking distances from 30 m.p.h. ...
-1
votes
1answer
34 views

What is a nice, clean proof to show that a fair coin toss satisfies axioms of probability?

If we assume H=Heads T=Tails and we're dealing with a fair coin what is a good way we can show that Kolmogorov Axiom has been satisfied?
-1
votes
0answers
13 views

Expected Value of covariance

You have an urn with balls that are either red or blue, and striped or not. What is the probability distribution that describes the number of blue balls drawn given the number of striped ones drawn? ...
0
votes
1answer
19 views

Having an independent event with animals

In a building for 24 apartments. It is known that there is only one dog in 8 apartments and a single cat in 6 apartments. How many apartments must have cat and dog for events "have dog" and " have ...
2
votes
2answers
40 views

Confusing probability problems based on product rule and combinations

I am going thru probability exercise. Faced first problem: Book Q1. Ten tickets are numbered 1,2,3,...,10. Six tickets are selected at random one at a time with replacement. What is the ...
1
vote
0answers
23 views

Normal and poissonian probability problems

I am working on a problem with a normal probability distribution but I am unsure of the results I calculated the probability asked for but still hesitate regarding the output and especially the first ...
1
vote
1answer
42 views

Probability of no 6 or no 5 when dice is rolled n times

Can anyone guide me in the general direction of the answer to the following: A die is rolled $n$ times $$A = \text{no $6$s}$$ $$B = \text{no $5$s}$$ $$P(A\cup B) = \;?$$ I am first finding $P(A)$ ...
0
votes
2answers
42 views

Independence between conditional expectations

Suppose $(\Omega, F, P)$ is a sample space, $X$ and $Y$ random variables, and $N$ and $M$ sub sigma algebras of $F$. I know that $E(X\mid N)$ and $E(X\mid\{\emptyset, \Omega\})$ are independent. ...
1
vote
1answer
42 views

The ant is moving through the coordinate system, Started at $(0,0)$ to $(4,4)$. What is the probability that the ant will find food at $(3,2)$?

The path to the $(3,2)$ is $3+2 \choose 3$ or $3+2 \choose 2$. Total path is $4+4 \choose 4$ And the probability is : $ \frac{3+2 \choose 3}{4+4 \choose 4}$ = $ \frac{5 \choose 3}{8 \choose 4}$ = ...
-1
votes
1answer
16 views

Length of random interval in $[0,1]$ that contains point $x$ [on hold]

The interval $(0,1)$ is divided by a point uniformly distributed in the interval $(0,1)$. Given $x \in (0,1)$, find the average length subrange which contains the point x. Show that this average is ...
0
votes
2answers
24 views

How to prove expectation exists (or improper integral converges)

How can I prove this improper integral converges, or give a counterexample? $$\int_{-\infty}^{\infty}x^n p(x)dx$$ where the only thing we know about $p(x)$ is $$\int_{-\infty}^{\infty}p(x)dx = 1 $$ ...
-1
votes
1answer
22 views

Are the converses of the following special cases of conditional expectation also true?

Let $X$ be a random variable, and $N$ be a sub sigma algebra of the underlyign sigma algebra of the sample space. if $X$ is in $L^1$ and measurable wrt $N$, then $E(X|N)=X$ a.e.. Is it true that ...
0
votes
1answer
21 views

Justification for Interchange of integral and sum

Let $\mu$ be a probability measure and $t\in\mathbb{R}$. I would like to write this equality $$\int_{\mathbb{R}}e^{ixt}d\mu(x)=\sum_{n\geq0}\frac{(it)^{n}}{n!}\int_{\mathbb{R}}x^{n}d\mu(x).$$ This is ...
2
votes
2answers
28 views

A conjecture about generating algebras on a probability space

Suppose that $(X,\mathscr F,\mathbb P)$ is a probability space. Let $\mathscr E\subseteq\mathscr F$ be an algebra (i.e., it is a non-empty collection closed under complementation and finite unions) ...
0
votes
4answers
23 views

Probability of extracting twice same ticket out of 4 pcs

I have just extracted from 2 consecutive tries the same ticket out of 4. How do I calculate the probability of such an event?
1
vote
2answers
34 views

Taking a ball from an urn after passing a randomly chosen ball from another urm

An urn contains four blue balls and three white balls. A second urn contains five blue and four white balls. Pass up a ball from the first to the second urn and then extracted a ball second urn. I ...
0
votes
0answers
16 views

probability of mutations occuring by chance mutually exclusively in cancer

I have a dataset that tells me if there are mutations in any of 500 genes in 100 cancer patients. Some patients have 0 mutations and some have >200. Genes generally work in networks, some of the genes ...
0
votes
0answers
56 views

From the binomial distribution

A single cell can either die, with probability $0.1$, or split into two cells, with probability $0.9$, producing a new generation of cells. Each cell in the new generation dies or splits into two ...
4
votes
2answers
50 views

Coin toss problem, get exactly 2 heads in 5 tosses

Suppose we toss a fair coin until we get exactly 2 heads. What is the probability that exactly 5 tosses are required? My try: We have to make sure that the first 4 tosses does not have 2 ...
2
votes
1answer
27 views

Birhdays: find the probabilities for the various configurations of the birthdays of 22 people

Let S,D,T,Q stand for simple,double,triple and quadruple, respectively: So, for example: the probabilities of 22 simple birthdays(22 person have birthdays in different days) are $ P(22S) = ...
0
votes
1answer
32 views

Probability and balls

An urn contains four blue balls and three white balls. A second urn contains five blue and four white balls. Pass up a ball from the first to the second urn and then extracted a ball second urn. How ...
-2
votes
1answer
24 views

How to arrange balls on a shelf without two similar colored balls being together? [on hold]

Given an infinite number of red and green balls,find the possible number of ways of arranging them on a shelf of size $n$ such that no two red balls are together.
0
votes
0answers
20 views

How to model time changing random variables

Lets say I have a random variable $X(t)$ which describes some unit of motion of a living organism and $X(t)$ is itself a timeseries since this unit of motion changes in time. I would like to be able ...
0
votes
3answers
31 views

What is the probability that after pulling out of a card deck 3 heart cards, that the 4th card will be also a heart?

What is the probability that after pulling out of a card deck 3 heart cards, that the 4th card will be also a heart? There are 52 cards in the deck and there is no replacement. $$P(4\text{th heart} | ...
0
votes
2answers
47 views

PIN password: possibilities with several users in an attempt.

I have a $4$-digit PIN and a list "user:pin". The possibility of guest the PIN's user is \frac{1}{10.000}. Example: user1:0001. But if I try 10.000 user at the same time what is the possiblity?. ...
0
votes
1answer
24 views

Binomial distribution in Probability and Statistics

Look at the binomial distribution with $n$ trials and probability $p$ of success on each trial. For what value of $k$ is $P(x=k)$ maximized? The mode of the distribution. Hint consider using ...
3
votes
2answers
43 views

Alternative interpretation of ball and urns problem

Suppose an urn has r red balls and b black balls. They are withdrawn one at a time at random until a total of k, k $\leq$ r, red balls have been withdrawn. Find the probability that a total of n balls ...
0
votes
3answers
43 views

In a game of Bridge, what is the probability that all 4 players are dealt 13 cards of the same suit?

I was asked this question by a student at my college, and I answered it like this: Since Bridge is played with 4 players, and there are 4 suits per deck of 52 cards, and assuming the deck is a fair, ...
0
votes
1answer
24 views

Coin flipping and probability

One in each two people launches three equilibrated coins. How likely is it that take the same number of heads??? guy $1$, just head $1/2 \cdot 1/2\cdot 1/2= 1/8$ guy $2$, just head $= 1/8$ But how ...
2
votes
1answer
62 views

Is it always true that $P(A \cap B) = P(A) + P(B) - P(A \cup B) $?

I saw this equation in a solution to a question, but maybe it's taken out of context. I didn't see this equation mentioned in the textbook. So is the above equation always true?
-1
votes
2answers
31 views

lamps and statistic [on hold]

I tried so hard this question, but I was not be able to answer it.... Could you help me to understand it? In a supermarket 2,000 lamps from three different factories A, B and C. The A produced 500 ...
0
votes
0answers
18 views

Max function as bounded functions

I have an algebra of bounded functions $A$ that contains the constant functions and is closed under uniform convergence. We also have that if $f \in A$ then $|f| \in A$. I'm trying to show that if $f, ...
0
votes
1answer
21 views

Say we have a double-decker Lazy Susan with two levels that can be turned independently. If we have n + k dishes in total, how many ways

Say we have a double-decker Lazy Susan with two levels that can be turned independently. If we have n + k dishes in total, how many ways is that solution is correct ???
1
vote
2answers
24 views

Solving for an expected value from discrete random variables

I'm having trouble seeing where I'm going wrong with a problem. The is the question: An urn contains 30 marbles of which 8 are black, 12 are red, and 10 are blue. Randomly, select four marbles ...
0
votes
1answer
30 views

A Characterization of the Strong Markov Property

I have a question concerning the strong Markov property: For a strong Markov process $(X_u)_{u\ge 0}$, a real time $t\in \mathbb{R}$ and an optional stopping time $T$ with $t< T$ \begin{align*} ...
-2
votes
1answer
25 views

Probability and Statistics - to understand expectation and variance better [on hold]

A strange clause in a version of Dungeons and Dragons says: roll a d6 (a six-sided die with faces from 1 to 6). If the value rolled is 3 or less, roll a d8 else roll a d10. Add the two values ...
4
votes
1answer
19 views

What is the probability that a customer will not use a credit card? Pays in cash or with a credit card?

So I'm doing some basic probability problems for homework, and we just recently went over the Inclusion-Exclusion prinicple, which I'm assuming this problem deals with, which is as follows. ...
-4
votes
1answer
27 views

STATISTICS AND PROBABILITY [on hold]

John and Isaac shot at a target. The probability that John hit the target is 1/4 and the probability that Isaac hit the target is 3/5. If they shot together, what is the probability that; A) both John ...
1
vote
2answers
34 views

Problem with injective functions on an explanation of the Birthday problem

The Wikipedia article on the Birthday problem gives an "abstract proof" of the problem, in which the birthday function $$ b:\mathcal{S} \mapsto \mathcal{B} $$ where $\mathcal{S}$ is the set of ...
5
votes
2answers
56 views

Proof that $2^n-(n+1) $ equations are necessary to establish the independence of n events.

Suppose $A_1,A_2,\cdots,A_n$ are $n$ events, we say that they are all independent if for all $\{i_1,\cdots, i_m\}\subset \{1,2,\cdots,n\}$(where $m\ge 2$), we have $$\mathrm{Pr}[A_{i_1}\cap ...
1
vote
2answers
28 views

Equivalence of Definitions of lim inf of Sequence of Sets

Prove : $\{w : w \in A_n \text{ for all $n$ except a finite number}\}= \bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}A_k$. I am trying to prove these two definitions are equivalent but I am having ...
-5
votes
4answers
70 views

Supervisor needs help. Is she really sick on Mondays? [on hold]

Employee has a total of 24 [full-day] absences over a year. She works four ten-hour days instead of five eight-hour days. Of the 24 absences, 13 have occurred on Mondays. I don't want to just sit ...