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
11 views

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

saw it in a solution to a question but maybe taken out of context. So is the above equation always true?
-1
votes
2answers
24 views

lamps and statistic

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
11 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
17 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
17 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
0answers
18 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
20 views

Probability and Statistics - to understand expectation and variance better

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
17 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
23 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
23 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 ...
3
votes
1answer
35 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
21 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
61 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
21 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
24 views

calculating compound probabiliy

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
23 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 ...
-2
votes
1answer
22 views

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

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: ...
-2
votes
0answers
40 views

How to write this in R? [on hold]

I want to write this in R: $$ \mbox{Pr}(X_1=x_1,\ldots,X_n=x_n)=\sum_{\lambda_1}\ldots\sum_{\lambda_n}\prod_{i=1}^n\left(\frac{\lambda_{i}^{x_i}}{x_i!}\right)e^{-\lambda_i}g(\lambda_i)$$ How can I ...
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
16 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
31 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
108 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
30 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
52 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
23 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
18 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
44 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 ...
1
vote
1answer
18 views

Square with different densities. Computing probability. [on hold]

I have a question about computing P(Y<0.5). Inside this square [-1,1] x [-1,1] we have different density function f(x,y). We can do it directly by counting area and it is 0.75. Because 4 is area ...
1
vote
1answer
32 views

Likelihood at least 2 out of $n$ numbers are visible to each other in $\mathbb{Z}^n$

Two points in $ \mathbb{Z}^n $ are said to be visible to each other, if they can be connected by a straight line, which doesn't intersect any points of $ \mathbb{Z}^n $ In Apostol's book "An ...
0
votes
1answer
35 views

What's the summary probability of an event if it increases over time?

I'm having trouble calculating this one. Say there are two steps an event occurs with certain probability: 60% 70% What is the probability that an event occurs by the time second step is reached? ...
0
votes
0answers
12 views

Is correct my Procedure about Joint Distribution for independent random variables

$ y_i, i=1,2...n$ are random variables are linearly independent For $y_i \sim Ber(p)$ $(p^{x_1}q^{1-x_1})(p^{x_2}q^{1-x_2})\bullet \bullet \bullet (p^{x_n}q^{1-x_n})$ ...
-2
votes
0answers
35 views

Counting math problems [on hold]

1) Ann, Bobby, and Cece are randomly placed in a line with 26 people total. What is the probability that Ann is to the left of Bobby, and Bobby is to the left of Cece? Express your answer as a common ...
-3
votes
1answer
43 views

What are the expectations of $1/X$ and $1/(1-X)$ if x has a Dirichlet distribution? [on hold]

What are the expectations of $X^{-1}$ and $(1-X)^{-1}$ if $X$ has a Dirichlet distribution?
1
vote
1answer
39 views

Show $X$ and $Y$ are independent if we assume that $E[XY] = E[X] E[Y] $

Assume that $$E[XY] = E[X]E[Y]$$ Let $X$ and $Y$ be random variables taking two different values $a,b \in \mathbb{R}$. Show that X and Y are independent. Note: I've spent a long time on this ...
0
votes
1answer
16 views

Show that f is a density and find the corresponding cdf

$f(x) = \frac{(1+\alpha x)}{2} $ for $-1 \leq x \leq 1$ and $f(x) = 0$ otherwise, where $-1\leq \alpha \leq 1$. Show that $f$ is a density and find the cdf. I am mainly having trouble with finding ...
0
votes
0answers
24 views

Markov Property Definition

Let $(X_t)$ be a stochastic process on $(\Omega, \mathcal F, \{\mathcal F_t\}, \mathbb P)$. The typical definition of the Markov property is $\mathbf{P}(X_{t+s} \le x \, |\, \mathcal F_t) = ...
-1
votes
0answers
16 views

Collision detection for two moving objects

There are 2 objects $A$ and $B$. Both have 2 sensors. The sensors can measure a distance to another sensor. Let's say the sensors are $AF$ (front sensor for $A$), $AR$ (rear sensor for $A$), $BF$, ...
0
votes
1answer
7 views

Walking through the reduction of a cumulative probability function to a polynomial

Setup Define $P(p)$ as follows: $$ P(p) = \sum_{N_1-\phi \cdot N_2 \geq \theta} {n_1 \choose N_1} {n_2 \choose N_2} p^{N_1 + N_2}q^{n_1 + n_2 - N_1 - N_2}. $$ Here, $$ q = 1 - p. $$ The sum is ...
-3
votes
1answer
33 views

Cats and Dogs = Idenpedent events [on hold]

I did not get this question. Could you explain it to me? 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 ...
-4
votes
0answers
17 views

Probabilityorchance [on hold]

i have a list of 9 people 1 of whom will be elected by a group of 65 people. The rules are this. Each person votes for three people. Each ballot must contain 3 different names. Each person complies ...
0
votes
0answers
14 views

Questions about solution to finding solution to mode of a binomial distribution

So i read over the solution presented by Andre Nicolas: finding mode in Binomial distribution But i have a few questions about the whole thing: 1) why did he set the ratio as $\frac{a_{k+1}}{a_k}$? ...
0
votes
2answers
29 views

Let $N$~Pois$(\lambda)$, $X|(N=n)$~Bin$(N,p)$, $Y=N-X$. Show $X$, $Y$ are independent and Poisson with parameters $\lambda p$ and $\lambda (1-p)$.

Any direction on this problem would be much appreciated. So far I know the joint distribution of $X$ and $Y$ is $\begin{align} \mathsf P(X=x, Y=y) & = \mathsf P(X=x, N-X=y) \\ & = \mathsf ...
0
votes
1answer
47 views

Finding patterns in seemingly arbitrary pairs of numbers

I don't work (directly) in mathematics (I'm a programmer), but I see numbers every day. Today I came across an issue where some totals were off, and was sent a list of the last 9 examples of the ...
-1
votes
0answers
15 views

Probability Data Management [on hold]

A bag contains 54 black marbles and 63 white marbles. Use Pascal’s Triangle to determine how many combinations and how many permutations are possible if 7 marbles are drawn out of the bag.
2
votes
3answers
65 views

How do I find the constant C?

Consider a random experiment with a sample space $$S=\{1,2,3,⋯\}$$. Suppose that we know: $$P(k) = P({k}) = \frac {c}{3^k}$$ for $k=1,2,⋯,$ where c is a constant. Find c. Find $P(\{2,4,6\})$. Find ...
-2
votes
1answer
64 views

Conditional distribution of mixed process

Let $Y$ be a random variable such that: $$Y \sim \begin{cases} \operatorname{Poiss}(\lambda_0), & x= 0 \\ \operatorname{Poiss}(\lambda_1), & x=1 \\ \end{cases} $$ where ...