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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0
votes
2answers
19 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 $$\int_{-\infty}^{\infty}p(x)dx = 1 $$ ...
0
votes
1answer
17 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
12 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 ...
1
vote
2answers
21 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
19 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
25 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
11 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
32 views

From the binomial distribution

A cell can either die or live with probability $0.1$ and $0.9$ respectively from one generation to next. How do we find the probability mass function for the number of cells in any generation?
4
votes
2answers
48 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
23 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
26 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
0answers
22 views

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

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
19 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
28 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
46 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
22 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 ...
2
votes
2answers
36 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
41 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
22 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
56 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
28 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
17 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
23 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
26 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
22 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
18 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
26 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
29 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 ...
4
votes
2answers
49 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
23 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
65 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 ...
0
votes
0answers
22 views

Proof that a PGM gives a probability distribution

A probabilistic graphical model defines a joint probability as: $$\mathbf{P} (X_1 \in A_1, \ldots, X_k \in A_k) = \prod_{i = 1}^k \mathbf{P} (X_i \in A_i \mid (X_j \in A_j)_{j \in \text{parents} ...
0
votes
0answers
26 views

calculating compound probabiliy [on hold]

If I have 100 people and they each have a choice of 500 sweets, how can I calculate the probability of 2 or 3 sweets being chosen mutually exclusively? For example (i'll give the sweets letter codes) ...
0
votes
0answers
26 views

How to prove the following

Let $\mathbf{A}\in\mathbb{R}^{p\times n} (n\ge p)$ be a positive definite symmetric matrix having a Wishart distribution with mean $\mathbf{0}$ and covariance $\boldsymbol\Sigma\otimes \mathbf{I}$. ...
2
votes
1answer
51 views

Numbers $\alpha$ and $\beta$ are selected from interval $[0,1]$. What is the probability that $x^2+\alpha x + \beta ^2=0$ has real roots?

I know that discriminant must be greater than zero , so we have : $\alpha ^2-4\beta^2\geq 0$ $\alpha^2\geq4\beta^2$ $\alpha\geq 2\beta$ We draw a function $\alpha - 2\beta = 0 $ and our condition ...
-3
votes
1answer
24 views

Probability that 2 people share the same Birth date, month, and day of the week. [on hold]

I just found out that my business partner and I were both born on a Friday, May 13th. What are the chances of that ? Considering random selection of two people. Curious ! Thanks very much, ...
-4
votes
0answers
17 views

Cumulative failure rate for hard drives [on hold]

Google have reported on the average failure rates of population of hard drives over time. They report the following statistics (approximated from their graph) for average failure rate: ...
1
vote
1answer
24 views

probability: A' ∩ B' ∩ C and (A' ∩ B')U C

how can I find the probability of the two events: $P(A)=0.22, P(B)=0.25, P(C)=0.28, P(A ∩ B) =0.11 P(A ∩ C)=0.05 P(B ∩ C)=0.07 P(A ∩ B ∩ C) = 0.01$ 1) $A' ∩ B' ∩ C $ I know that $A' ∩ B' = (A \cup ...
1
vote
1answer
18 views

Probability or optimization

I have a problem with the following case $F$ and $G$ are distribution function on $x\in{[0,1]}$ and they have same mean $\mu$ I want to prove $\int_0^1 F(x)G(x)dx\geq(\mu-1)^2$
0
votes
1answer
20 views

Poisson sampling

Suppose I have a pdf $f(S)$. $f(S)$ describes the size of firms in the economy. Also define the Bernoulli variable $X_{f} \in \{0,1\}$ where $P(X_{f}=1)=g(S_{f})$ and $P(X_{f}=0)=1-g(S_{f})$. $S_{f}$ ...
0
votes
1answer
34 views

Permutation and combination/ probability [duplicate]

If you have 7 white socks and 9 black socks in a drawer, how many socks do you have to pull out blindly in order to ensure that you have a matching pair ?
5
votes
4answers
111 views

How many ways to write $2010$?

Let $ N$ be the number of ways to write $ 2010$ in the form $ 2010 = a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0$, where the $ a_i$'s are integers, and $ 0 \le a_i \le 99$. An example of ...
0
votes
0answers
34 views

What is the probability that from 23 people 2 people have their birthday on the same day?

What is the probability that from 23 at least people 2 people have their birthday on the same day. Assume that the year has 365 days and that all the birthday combinations have the same probability. ...
-1
votes
2answers
55 views

Expected value of a biased coin toss

Please help me to calculate expected value. Consider a biased coins such that the probability for tails is p and the probability for heads is 1-p. Coin tossing continued until the coin shows heads. ...
1
vote
0answers
26 views

Problem with statistics notation for a density function

I'm reading a paper about partitioning of driving data and producing synthetic driiving profiles and I'm uncapable of understanding some of its equations. Just to give an example, if we consider the ...
0
votes
0answers
22 views

how to determine presence of an event with a degree of confidence proportional to a set of observations and conditional probabilities

My probability theory has become a bit rusty and i can't seem to figure out how to determine the presence of a malfunction within a device given a set of observations displaying a certain phenomenon ...
2
votes
1answer
29 views

Random ants probability question

500 ants are randomly put on a 1-foot string (independent uniform distribution for each ant between 0 and 1). Each ant randomly moves toward on end of the string (equal probability to the left or the ...
2
votes
0answers
19 views

Generalised equation/Notation for writing down products of sets of combinations

I am trying to write a generalized equation to solve a fairly simple probability problem (c & k are constants) $$y_{1} = (1 - cx_{1})^k$$ $$y_{2} = \frac{(1 - cx_{2})^k - ...
2
votes
0answers
45 views

Transformation of probability density function

I'd like to compute the pdf of $w= g_1(x) = \frac{x}{1+e^{-x}}$ in dependence of the density $f_x(x)$ with domain $x>0$. As I was not able to write the inverse function of $g_1(x)$, I tried the ...