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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2
votes
1answer
40 views

Most likely order of independent normal random events

The problem I have is, given $n$ independent normal distributions describing the times that $n$ random events occur at, what is the most likely order that they will occur in? This questions follows ...
2
votes
1answer
72 views

Notation i.i.d sample

I am learning measure theory and sometimes I am not sure if I am using the correct notations, especially with respect to distributions of random variables. In the following I try to formulate the ...
2
votes
0answers
30 views

Indicator Functions properties in Probability

This is probably quite a straightforward question but I just thought I should double check my working. Say for instance we have two fixed values $K_{1}$ and $K_{2}$ such that $K_{2} > K_{1}$ and ...
-2
votes
2answers
46 views

If $1$ boy and $1$ girl are selected at random, what is the probability that both of them scores $90$ marks above? [closed]

There are $25$ boys and $15$ girls in a class. $5$ boys and $1$ girl has got $90$ marks above in exam. If $1$ boy and $1$ girl are selected at random, what is the probability that both of them ...
0
votes
3answers
67 views

$Y$ can only take on $\{−1, 0, 1\}$. The expected value of $Y$ is $0$ and its variance is $1/2$. Find the probability distribution of $Y$.

How would one approach this question? A random variable Y can only take values in $\{−1, 0, 1\}$. The expected value of $Y$ is $0$ and its variance is $1/2$. Find the probability distribution of $Y$. ...
3
votes
2answers
50 views

Question on Probability. What is my Z-Score 20?

I have this homework but I am getting a z-score of 20? Why? Is my calculation wrong? The question is Without assuming that the diameters of apple pies are distributed according to the normal ...
1
vote
1answer
28 views

Combinatoric/Partitioning exercise Stardew Valley

I am playing a popular computer game called Stardew Valley. In the game, you can take a fruit/vegetable put it into a seed maker to get more seeds. If I put 1 tomato into the seedmaker, itt can ...
1
vote
0answers
40 views

Evaluate the combination of $\sum\limits_{j=0}^{{\lceil} \frac{k}{2} {\rceil}}\binom{N-k}{j}$

Can any one help me please to get the approximate result of this combination problem using asymptotic notation: $$ \sum\limits_{j=0}^{{\lceil} \frac{k}{2} {\rceil}}\binom{N-k}{j} $$ Thanks
1
vote
1answer
73 views

Recursive calculation - How would it be started?

First off, let me assure you that this is my very last resource. I have tried everything, because I understand that posting a question that needs a hyperlink to be understood is annoying. But I would ...
1
vote
1answer
23 views

Fluctuations in estimator of $\min\{p,1-p\}$

Let $X_1,\ldots,X_n$ be i.i.d. Bernoulli with some parameter $1/2$. Let $\bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i$. I am trying to show $$\mathbb{E} \min\{\bar{X}_n,1-\bar{X}_n\} \ge \frac{1}{2} - C ...
0
votes
1answer
38 views

What are the odds of guessing 3 out of 4 numbers of a pin correctly, out of order?

So the probability of guessing a four digit pin is $\frac{1}{10000}$, given that the possible combinations are $0000$-$9999$. Guessing one digit (in order) correctly is $\frac{1}{10}$, the second ...
2
votes
1answer
19 views

Max Likelihood Examples, Stuck in Calculation [closed]

We get samples 2,4,8,16 be random instances that get from distribution with following PDF. maximum likelihood estimation of $ (\alpha, \sigma) $ is : $ \frac {2}{3 ln 2}, 2$. $ f_{\alpha, ...
0
votes
1answer
12 views

Two exponentially distributed random variables w/ different intensity. Which is more probable to take?

Let's say I have two types of light bulbs, A which has $E(A)=100$ hours of lifetime, and B which has $E(B)=200$. I have three of type A and one of type B. I randomly use one of the four, and after 200 ...
0
votes
0answers
11 views

K-wise identical marginal distributions

Suppose I have two joint distributions described by the two sequences of random variables,$X_1, \ldots, X_n$; $Y_1, \ldots, Y_n$. Is there a name/theory/reference for when these two distributions ...
-1
votes
0answers
35 views

The probability density function of the product of independent exponentially distributed RVs

When $X$ ~ exponential distribution(10) and $Y$ ~ exponential distribution(15), and they are independent, find the probability density function of $Z = XY$ I just took an exam for probability ...
0
votes
1answer
31 views

Find a Recursion Formula with boundary conditions

A game is played as follows. Coins with value $x_1, . . . , x_n$ are laid on a table in a row. Two players alternate turns taking a coin from either end of what’s left of the row of coins until the ...
0
votes
2answers
43 views

Probability of a right angled triangle with sides a+b=200 having hypotenuse ≥ 160

QUESTION: A $200\, cm$ long staff breaks in two at a random point. The two parts becomes the right sides of a right angled triangle. What is the probability of the hypotenuse being at least $160\,cm$? ...
2
votes
1answer
30 views

Probability of selecting a jury

Does anyone know how to find the probability of selecting a jury of $12$ people ($6$ men and $6$ women) out of an initial group of $18$ people ($6$ men and $12$ women)? With my knowledge of ...
0
votes
1answer
34 views

Valuing a game using recursion

An urn contains 4 marbles: 2 red and 2 green. You extract them one by one without replacement. If you extract a green marble, I pay you 1 dollar; if you extract a red marble, you pay me $1.25. You can ...
1
vote
1answer
36 views

Probability - Definition

Probability = $\dfrac{\text{Number of favorable outcomes}}{\text{ total number of outcome}}$ The above definition is taken from my textbook and also available in many websites. But, after ...
2
votes
1answer
40 views

How to calculate expected time of successive events

First time posting here. Apologies if this is too basic on not properly asked. I am trying to solve a puzzle programmatically an the problem is as follows: We have several individuals who will go ...
2
votes
1answer
21 views

Understanding statistical independence of events using a relative frequency interpretation

This is what I've read in my textbook: "If $n_A$ and $n_B$ are the number of times the independent events $A$ and $B$ have occurred, then we expect that the ratio $\frac{n_{AB}}{n_A}$ (num. of times ...
5
votes
6answers
761 views

Chances of being picked last in a hat [duplicate]

A couple of friends and I are struggling with coming up with an answer to this question. It's seemingly simple but I need a little help. 6 people put their name into a hat. One by one, names ...
6
votes
0answers
61 views

Progressive Dice Game

You have a fair, regular 6-sided dice. The game is played for $n$ turns. Each turn you make a roll and gain that many points the rolled side is showing, then do one of the following: ...
0
votes
1answer
20 views

Sum Before Crossing Value x

I'm having trouble answering the following question: Let $X_1$, $X_2$, .... be uniform [0,1] iid RVs. Define: $t(x) = min(n>0:X_1+X_2+....X_n > x)$. Find $P(t(x)>2).$ To get an idea of ...
3
votes
0answers
44 views

How to punish a riffle shuffle?

It is common knowledge that for a deck to be considered randomly sorted, at least seven riffle shuffles should be used. However, in my experience, very few people will take the time to complete seven ...
5
votes
2answers
83 views

Mario Party 3 Mini-game Probability Question

I have a question about a mini-game in Mario Party 3. I have extracted the mathematical information from the game below. Setup: Four players $A,B,C$, and $D$ line up in some order. There are $12$ ...
0
votes
0answers
27 views

Expected value of maximum likelihood estimator of a Bernoulli random variable

While reading the text from Keith H. Thompson on the Estimation of the Proportion of Vectors in a Natural Population of Insects, I came across the following part where I don't understand everything. ...
0
votes
1answer
22 views

Any way to calculate chances of getting “n” hits when rolling “x” die (hit is when I roll more than “y”)?

First, let me preface - I saw similar questions already, but to be honest, I didn't understand the answers, or couldn't understand how to convert the answer for given question to my problem. My ...
1
vote
0answers
17 views

Distribution of Normal Mixture of Uniforms

Let's say I know that \begin{align*} X \mid \mu,\sigma & \sim \mathcal{N}(\mu, \sigma^2), \\ \mu & \sim Unif(a,b), \\ \sigma & \sim Unif(c,d). \end{align*} I'd like to know the marginal ...
1
vote
1answer
16 views

How to find the variance of a normal distribution?

X has normal distribution with the expected value of 70 and variance of σ. It is known that $P(67.36\le X \le 72.64) = 0.34$ find σ So if I understand this right we know that ...
0
votes
1answer
111 views

How to apply the example of cars passing through a road to the Poisson distribution?

Cars pass through a road junction according to a Poisson distribution. An average of 5 cars per minute pass through the junction. a) What is the probability that exactly one car passes ...
-2
votes
0answers
22 views

The conditional probability density function with a specific condition [on hold]

Assume the following discrete time model: $x(t+1)=Ax(t)+w(t)$ where $w(t)$ is zero mean, iid white noise with bounded covariance matrix $Q$. Let $s=x(t)+x(t-1)+x(t-2)$. How I can find ...
-1
votes
0answers
30 views

Stochastics Process [closed]

There are two transatlantic cables each of which can handle one telegraph message at a time. The time-to-breakdown for each has the same exponential distribution with parameter λ. The time to repair ...
0
votes
1answer
13 views

Correlated samples due to Metropolis algorithm

The Wikipedia article about the Metropolis algorith notes one disadvantage as follows: The samples are correlated. Even though over the long term they do correctly follow P(x), a set of nearby ...
6
votes
4answers
895 views

What is meant by “identically distributed”?

When two distributions have the same variance and shape, do we call them identically distributed regardless of their mean (as this is usually a location parameter)? Thanks
3
votes
0answers
70 views

Suppose $E[X_1] <\infty$. Show that $lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s.

Let $X_1,X_2,X_3,...$ be i.i.d. with $P(X_1 >0)=1$. Define $S_n =\Sigma_{i=1}^{n} X_i$. (a) Suppose $\mathbb{E}[X_1] <\infty$. Show that $\lim_{n\rightarrow \infty} \frac{X_n}{S_n}=0$ a.s. I ...
0
votes
3answers
24 views

experiments and probabilities

If I perform an experiment with 60 trials and I decide to triple the trials to 180, why would i get a probability closer to the true probability with 180 trials compared to 60 trials? I feel like ...
0
votes
0answers
32 views

Fair Coins and Method of Moments [closed]

If I have a card (red/black) that is dealt N times during three trials - first trial yields 4 red, second and third trials yield 6 red, is it more likely that N (the number of deals) is 8 or 10 and ...
0
votes
2answers
30 views

$\mathcal{P}$ stochastic matrix. If there is $k > 0$ st $\mathcal{P}^k(j, i) > 0$, then there is $r \leq (n-1)$ st $\mathcal{P}^r(j, i) > 0$

Let $\mathcal{P}$ be stochastic matrix of order n. If there is $k > 0$ such that $\mathcal{P}^k(j, i) > 0$, then there is $r \leq (n-1)$ such that $\mathcal{P}^r(j, i) > 0$. My attempt: ...
3
votes
1answer
40 views

Probability that two random walks on $\mathbb{Z}^2$ meet at the origin

Suppose $X,Y$ are symmetric, independent random walks on the lattice $\mathbb{Z}^2$. I am trying to find the probability: $$\mathbb{P}\big(X_n=Y_n=(0,0)\;\text{for ...
1
vote
0answers
12 views

Sampling with Clusters

Suppose we have a population of size $N$ where each individual is represented by a vector of the form, $v_i=(\lambda_1^{(i)},...,\lambda_s^{(i)})$ where $\lambda_k^{(i)}$ are finite discrete ...
0
votes
0answers
17 views

non square transformation of random variables

Let $x_0$ and $w_0$ be independent random variables and let $x_1$ be related to them by $x_1 = f(x_0, w_0)$. I want to find the joint density of $x_1, x_0, w_0$. The transformation I am interested ...
0
votes
1answer
26 views

Proof of conditional probability formula [closed]

Knowing only this: $P(A\mid B) = \frac{P(A\cap B)}{P(B)}$ How can i proof this? $P(A\cap C\mid B) = P(A\mid C\cap B).P(C\mid B)$
-1
votes
0answers
58 views

How to solve this probability question?

From a group of 50 people asked to pick a number between 1 and 100 (both inclusive), what is the probability that at least half of them will pick a single digit number? So far all I could come up ...
0
votes
1answer
57 views

A Question Regarding Markov Chains and Ergodicity

Suppose the Markov chain with Probability Transition Matrix, $P$ = ($p{_x}{_y}$) is ergodic and $p{_m}(x, y) > 0$ for all states $x$ and $y$. If $n ≥ m$, show that $p_n(x, y) > 0$ for all ...
1
vote
1answer
30 views

random variable probability problem

I am trying to find the answer to a mathematical probability problem. let a box contain $5$ balls : $2$ balls white, $2$ balls green, and $1$ red ball (we can't differentiate between the balls by ...
0
votes
0answers
33 views

How it is shown by the following integral?

Example: Ornstein-Uhlenbeck Process. Let $ dx=-\eta xdt+\sigma dz $ be an Ornstein-Uhlenbeck Process Write the moment-generating function for $x(t)$ as $$ M(θ,t)≡E(e^{-θx})=∫_\infty^∞ ...
1
vote
0answers
35 views

12 numbered pigeonholes and balls [duplicate]

This problem was inspired by this James Randi challenge. Given $12$ numbered ($1$ to $12$) pigeonholes and $12$ numbered balls (also from $1$ to $12$); what is the probability that a random ...
-2
votes
1answer
48 views

Probability of n-dimensional spheres in n-dimensional cubes?

Suppose we have a unit disk inside of a unit square, both centered at the origin. Then, we assign a random point $p$ to the space inside the unit square, and it may or may not be in the circle. If ...