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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35 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 ...
-1
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0answers
16 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. ...
0
votes
1answer
31 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
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0answers
17 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
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0answers
9 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
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1answer
25 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 ...
5
votes
1answer
36 views

Inference on $P\left(\left.\sum_{i=1}^{N}X_{i}\right|\sum_{i=1}^{N}X_{i}^{2}\right)$ when $X_{i}\sim\mathcal{N}\left(0,1\right)$?

Let $$X_{i}\overset{i.i.d}{\sim}\mathcal{N}\left(0,1\right)$$ Hence: $$\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)$$ and $$\sum_{i=1}^{N}X_{i}^{2}\sim\chi^{2}\left(N\right)$$ What ...
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0answers
27 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 $f_x(x)$. As I was not able to write the inverse function of $g_1(x)$, I tried the following approach: I ...
1
vote
1answer
16 views

Square with different densities. Computing probability.

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
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1answer
27 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 ...
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1answer
29 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
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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
33 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
40 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
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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
15 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
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0answers
19 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) = ...
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0answers
15 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
32 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
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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
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0answers
13 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
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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
45 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 ...
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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
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3answers
64 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
50 views

Conditional distribution of mixed process

$$ N(t)=(1-B).N_0(t)+B.N_1(t), \quad \quad \text{where B is Bernoulli($p$), $N_0(t) \sim \operatorname{Poiss}(\lambda_0 t)$ and $N_1(t) \sim \operatorname{Poiss}(\lambda_1 t)$}. $$ I suspect that ...
-1
votes
2answers
42 views

PDF of $Y=\min(0,X)$ when PDF of $X$ is $\frac34(1-x^2)$ on $(-1,1)$

Let $X$ be a random variable with density $f(x) = (3/4) (1-x^2).$ Range is $-1 < x < 1.$ I have to find probability distribution of $Y = \min(0,X).$ I know that distribution function could be ...
3
votes
1answer
26 views

Connecting noodles probability question

I don't know how to solve this. You have 100 noodles in your soup bowl. Being blindfolded, you are told to take two ends of some noodles (each end of any noodle has the same probability of being ...
4
votes
0answers
24 views

Given two sets of $100$ samples of $10$ items from a $1000$ item set, what is probability that the two sets have non-empty intersection

Suppose two people go grocery shopping $100$ times each. Each time, they pick $10$ items randomly from the $1000$ items at the store. As a result, each person has $100$ randomly chosen baskets of ...
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0answers
20 views

What is the Probability of drawing number less than 3 when a die os rolled? [on hold]

What is the Probability of getting number less than 3 when a die is rolled?
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0answers
19 views

Central Limits Theorem [on hold]

An instructor has 50 exams that will be graded in sequence. The times required to grade the 50 exams are independent, with a common distribution that has mean 20 minutes and standard deviation 4 ...
1
vote
1answer
17 views

Calculating the probability of a set of numbers appearing in a randomly-generated 3x3 grid

A challenge within a PC game I play features a 3x3 grid which contains all numbers from $1$ to $9$ in a random order. For example, this may be what a randomly-generated 3x3 grid looks like: ...
2
votes
3answers
41 views

Inductively defined random variables

Let $X_0=1$, define $X_n$ inductively by declaring that $X_{n+1}$ is uniformly distributed over $(0,X_n)$. Now I can't understand how does $X_{n}$ gets defined. If someone would just derive the ...
0
votes
2answers
46 views

'Probability method' - to what extent is it an actual proof?

Consider this: $\frac { C^{2n}_n } {2^{2n}} = \mathbb P (A) $ where A= { equal number of heads and tails in $2n$ throws of a fair coin } therefore the following assertion is true: $\forall n \ge ...
2
votes
2answers
33 views

What are the odds of spinning matching items in a slot machine?

Lets say we have a slot machine with $5$ reels. Each reel has $5$ different items on it. What are the odds of spinning $2, 3, 4$ and $5$ matching items? As I understand the probability of rolling a ...
3
votes
3answers
32 views

Combinatorics question on group of people making separate groups

If there are $9$ people, and $2$ groups get formed, one with $3$ people and one with $6$ people (at random), what is the probability that $2$ people, John and James, will end up in the same group? ...
0
votes
1answer
38 views

Probability of Team A winning where a draw is not allowed

I have the probabilities for a range of final scores for a sports team A and also for a sports team B. I assume that these probabilities are fixed and not affected by outside factors including the ...
0
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0answers
45 views

Additional Problems about conditional expectations

When I was a student, I was able to solve these problems, but now I can't because I forgot some important things. If you show me the solutions to at least two of them, I am sure, I will refresh it all ...
0
votes
3answers
51 views

Prove that if $A$ is independent of $B$, $A$ is independent of $C$, then $A$ is independent of $B\cup C$.

Prove that if $A$ is independent of $B$, $A$ is independent of $C$, then $A$ is independent of $B\cup C$. $\mathbb{P}(A)\mathbb{P}(B)=\mathbb{P}(AB)$ $\mathbb{P}(A)\mathbb{P}(C)=\mathbb{P}(AC)$ So ...
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0answers
33 views

Four Conditional Expectation Problems

I can't solve these problems and I will be very grateful to you for your help: 1) The random variables $X$ and $Y$ are independent and both have a uniform distribution $U([-1,1])$. Find ...
0
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0answers
35 views

Bookmaker's odds

Suppose a match can be completed in three ways: win, loose or draw. A bookmaker provide the following coefficients (including spread) for each case respectivelly: $c_1, c_2, c_3$. That is, if a player ...
1
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0answers
36 views

Biased Random Walk with Variable Probability

Consider a random walk in which the probability to move forward in time $t$ is $p_t$ and the probability to move backward is $q_t=1-p_t$ with $p_t<q_t$ with $p_t<p_{t+1}$ and $q_t>q_{t+1}$. ...
0
votes
2answers
24 views

Simple sequence of experiments

If I have an experiment with $\frac{1}{2}$ probability of success, how many times I have to run as to be 99.9% sure I am successful? Is someone please able to explain how to resolve this step-wise? ...
0
votes
1answer
15 views

How to determine if events are mutually independent.

Consider the experiment of tossing three coins. Let A be an event of getting head on the first coin, B be an event of getting tail on the second coin, and C be an event of getting at least two heads. ...
0
votes
3answers
29 views

Uniform PDF for continuous variable, why does the probability values increase to 1, when its normalized?

Consider a "spinner": an object like an unmagnetized compass needle that can pivots freely around an axis, and is stable pointing in any direction. You give it a spin and see where it comes to rest, ...
1
vote
2answers
38 views

Fundamentals of Probability

Suppose I have two boxes , containing white and black balls. In the first one , we have 8 white and 6 black balls. In the second one , we have 4 white and 7 black balls. Now if one ball is drawn at ...
0
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2answers
28 views

Classical Probability

There are three buttons which are painted red on one side and white on the other. If we tosses the buttons into the air, calculate the probability that all three come up the same color. Remarks: A ...
0
votes
1answer
14 views

inequality probability between order statistics of two independent distribution

Suppose we have two independent distributions $F_1$ and $F_2$ and from each distribution, we draw $k$ variables. Let us represent the $k$ i.i.d. variables from $F_1$ as $\{X_1, X_2, \ldots, X_k\}$. ...
0
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0answers
35 views

calculate the cdf

The continuous random variable $R$ has the following probability density function on the sample space $−1 \le r \le 1$, $$f(r)=\begin{cases} \frac{1}{4} \hspace{.5cm} \text{for} \hspace{.5cm} −1 ≤ r ≤ ...