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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3
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1answer
26 views

Probability question from GRE subject test

I ran across the following problem while practicing for the GRE math subject test: Suppose $X$ is a discrete random variable on the set of positive integers such that for each positive integer $n$, ...
-1
votes
0answers
27 views

How to calculate $P(X_1 < X_2 < X_3…X_n ) $

Could you please help with the following problem i am having- I need to calculate the probability of $X_1$ (randomly selected discrete value between $a$ and $b$) being smaller then $X_2$ (randomly ...
1
vote
2answers
27 views

Find the following probability

A bowl contains 16 chips, of which 6 are red, 7 are white and 3 are blue. If four chips are taken at random and without replacement, find the probability that there is at least 1 chip of each colour. ...
2
votes
1answer
36 views

Probability distribution of number of waiting customers in front of a counter

The number of customers arriving at a bank counter is in accordance with a Poisson distribution with mean rate of 5 customers in 3 minutes. Service time at the counter follows exponential distribution ...
0
votes
0answers
24 views

Show that the following set function is not a probability set function

If the sample space is $\mathfrak{C} = \{c : -\infty < c < \infty\}$ and if $C \subset \mathfrak{C}$ is a set for which the integral $\int\limits_C e^{-|x|}dx$ exists, show that this set ...
2
votes
2answers
15 views

Distribution of a product of Multinomials

Consider the following: $(X_1, X_2, X_3, X_4) \sim \mathrm{Multinomial} (n,\mathbf{p})$ where $\mathbf{p} = (p_1,p_2,p_3,p_4)$. I would like to find the distribution of $X_1 X_4$, or at least know ...
1
vote
1answer
28 views

Definition of standard deviation and $l_2$

If we denote the mean as $\mu$, then the standard deviation is: $$\sigma\equiv\left(\sum_{x\in X}{p(x)(x-\mu)^2}\right)^\frac{1}{2}$$ In other words, $\sigma$ is the average $l_2$ distance from $\mu$. ...
3
votes
1answer
40 views

How to take into account uncertainty on number of events

Suppose I generate a set of events $X_{i}$ for $i = 1,2 \dots N$ and suppose every event is either a success or a failure, ie. $X_{i} = 0, 1$. If $N$ is fixed, the MLE for the probability of success ...
0
votes
0answers
24 views

Confused about definition of absorption probability

My confusion can probably most easily be explained with an example. Consider the following one step transition matrix : $$ P=\matrix{% & 0 & 1 & 2 & 3 & 4 \\ 0 & ...
1
vote
1answer
31 views

“Time until arrival/departure” in a Poisson process…

Customers are served at a bank with the following process. While there is at most one customer in the bank, there will be only one person teller at a window. If a second customer comes into the ...
1
vote
2answers
21 views

For what fixed interest rates is a certain single-period, finite-state market arbitrage free?

A single period market with three states of nature $\omega_1$, $\omega_2$ and $\omega_3$ is given, in which a single asset is available, namely a stock that is worth $8$ units today, and whose payoff ...
1
vote
1answer
21 views

Rewriting probabilities as expectation

Consider the stopping time $\tau_a:=\lbrace{t>0| W_t >a\rbrace}$, where $W_t$ is a Brownian Motion. Define: $X_t:=W_{\tau_a+t}-W_{\tau_a}$. We have that $X_t$ is a Brownian Motion independent ...
-1
votes
1answer
38 views

A related problem regarding Normal Distribution (Continuous Probability) [on hold]

A circus performer who gets shot from a cannon is supposed to land in a safety net positioned at the other end of the arena. The distance he travels is normally distributed with a mean of 140 feet and ...
-1
votes
2answers
49 views

Question on Probability 11 [on hold]

The probability that $A,B$ and $C$ can solve a problem are ${4}\over{5}$,${2}\over{3}$ and ${3}\over{7}$ respectively . The probability of problem being solved by $A$ and $B$ is $8\over15$,$B$ and $C$ ...
0
votes
1answer
28 views

How to find the probability of declaring faulty

My question: Consider a company that assembles computers. The probability of a faulty assembly of any computer is $x$. The company therefore subjects each computer to a testing process. This testing ...
0
votes
1answer
38 views

Coin Toss Experiment

I conducted an experiment where I tossed a coin 100 times. I am assuming that the coin flips heads with a probability p=0.5. So that the coin is fair with a level of significance of 5%, I want to find ...
-1
votes
1answer
33 views

Probability for a game move [on hold]

There is $25\%$ chance that this skill activates the stun ability for each hit. The skill hits $4$ monsters and each monster is hit $4$ times , for a total of $16$ hits (each monster is hit $4$ times ...
3
votes
4answers
50 views

Binomial distribution, given the number of success, what is the expected total number of trials?

For a random variable that follows binomial distribution, $X|N=n\sim Binomial(n,p)$. What is the expectation of $N$ when we know the value of the random variable but don't know the total? ie. What is ...
2
votes
2answers
34 views

The Probability one Player will have more Kills than another based on a distribution of Kills? [on hold]

Alright, I'm definitely not a math guy so bare with me. I'll make this short and simple. I have a dataset of players and the # of kills (video game) they have per game. For instance, if there are 10 ...
2
votes
2answers
40 views

What conditional independence theorem is being used here

In stanford's machine learning lecture 1, linear regression is defined on page 11, section 3 as: For $i = 1, \ldots, m$, $y^{(i)} = \theta^T x^{(i)} + \epsilon^{(i)}$, where $\epsilon^{(i)}$ are IID ...
1
vote
1answer
21 views

Modelling a compound random variable from a Poisson process?

One other question I came across that I didn't quite understand. The number of forks that enter the sink follows a Poisson process with rate $λ= 200$ per month. Each fork which enters the sink ...
0
votes
2answers
36 views

Probability Uniform Distribution Set Up Integral

Consider a $1$ meter stick and suppose you break it into two pieces $X$ meters from the end, where $X \sim \operatorname{Unif}(0,1)$. What is the expected length of the longer piece (after it is ...
-2
votes
1answer
19 views

The probability that two randomly selected $2$ year old male feral cats will live to be $ 3$ years old is? [on hold]

The probability that a randomly selected $2$ year old male feral will live to be $3$ years old is $0.82666$. (a) what is the probability that two randomly selected $2$ year old male feral cats will ...
0
votes
0answers
24 views

Central Limit Theorem Absolute Inequality Equation

$\mu_N = 0.5$ (mean) $\sigma_N^2= \dfrac{1}{12N}$ (variance) $\sigma_N = \sqrt{\dfrac{1}{12N}}$ (standard deviation) For $N = 100$, what is $P(|X_N - (1/2)| > 0.025) =$ ? (i.e.using Central ...
0
votes
1answer
24 views

Suppose that $E$ and $F$ are two events? [on hold]

Suppose that $E$ and $F$ are two events and that $P(E\cap F)= 0.4$ and $P(E)= 0.8$. What is $P(F\mid E)$ ?
2
votes
5answers
97 views

Uncountable increasing family of $\sigma$-algebras

Could someone give an example of what an uncountable increasing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t\geq 0}$, $(\mathcal{F}_s \subset \mathcal{F}_t$ for $s<t)$ might look like? For ...
3
votes
2answers
38 views

Poisson process conditional probability problem

Penguins slide through a chute in a Poisson process at a rate of $2$ per minute. Each penguin has a $10$% chance of being an emperor penguin, independent of everything else. Given that $90$ ...
2
votes
2answers
28 views

Probability Distribution

I'm thinking about a set of n users on Facebook. Between each of the $\binom{n}{2}$ pairs of distinct friends, lets say an edge (indicating that the two people are friends) is independently present ...
0
votes
2answers
39 views

Probability and integration

Compute $E[e^{tX}]$ where $X ∼ \mathcal{N} (0, 1)$. [Hint: Complete the square in the exponent.] Do we set up the integral from $0$ to $1$? Then how do you solve this integral?
0
votes
0answers
26 views

Probability Problem - Using Bernoulli trials

This is a question on review for an upcoming test, any suggestions on solving or steps to find the solution would be greatly appreciated. "Angela thinks that if you spin a penny, it will land heads ...
1
vote
1answer
34 views

Continuous distribution and independence [on hold]

Problem: In a room, there are 4 boys from high income families, 6 girls from high income families and 6 boys from low income families. How many girls from low income families also need to be present ...
2
votes
0answers
45 views

Of strings and substrings: A problem of probability

Problem Let $\Sigma=\{a, b\}$. Let $\Sigma^*$ denote the Kleene star of $\Sigma$: \begin{equation*} \Sigma^* = \{\varepsilon, a, b, aa, ab, ba, bb, aaa, aab, \ldots\} \end{equation*} where ...
1
vote
0answers
24 views

Randomly searching a maze with a given probability distribution function

Consider a 2d maze in which there is one entrance and one exit. You are not a good maze solver so starting at the entrance, you try to find the entrance by naive random depth first search with ...
-4
votes
2answers
37 views

License Plate problem [on hold]

A license plate contains 7 characters (order matters). Each character may either be an upper-case letter A–Z or a number 0–9. How many license plates. . . (a) contain the string ABC? (b) have at ...
0
votes
0answers
17 views

Ratio check between two numbers dynamically scaling

Let's say we have one number wins: 8 and other number loses: 10 W/L ratio is 0,8. The chance to increase wins or loses is 50/50. Because of the large numbers law the ratio should get closer to 1,0 ...
1
vote
0answers
49 views

Probability and continuous distributions

Suppose that the daily consumption of pepsi in ounces is normally distributed with normal(13, 4) in ounces. The daily amount consumed is independent of other days except adjacent days where the ...
0
votes
1answer
25 views

Probability of winning once if there are 150 winners in 2000 tickets

I'm looking to enter a raffle. The are 150 winning tickets out of the 2000 given. I want to know my chances of winning at least once if I have 3 and 4 tickets. The initial answer I got was about ...
1
vote
1answer
25 views

Am I properly calculating this probability?

I'm trying to come up with a pure statistically probably of overlap for 3 non-exclusive groups by using Independence, so $P(A\cap B) = P(A)P(B)$ All groups make up part of the whole, but again, are ...
0
votes
1answer
33 views

Questions about Variance and Covariance [on hold]

I have a few questions about the linearity (or lack thereof) of covariance. Let $A_1, A_2.. An$ all be independent random variables that have the same mean $\mu$ and variance $\sigma^2$. (1) Would ...
-1
votes
1answer
22 views

Probability Question Likely Outcome of Event

The probability that you will win a certain game is 0.8. If you play the game 12 times, what is the probability that you will win at least 7 times?
4
votes
1answer
30 views

Brownian motion: Strong Markov versus translation invariance

In the proof of the reflection principle in Durrett's textbook (Probability: Theory and Examples (4e), Theorem 8.4.1, page 317), there's a step which I'm a little shaky on. Basically, this proof ...
1
vote
0answers
36 views

Probability and Uniform distribution lottery question

Suppose that a person has a lottery ticket from which she will win $X$ dollars, where $X \sim\mathrm{ Unif} (0,4)$. Suppose her utility function is $U(x) = x\alpha$ for $x \geq 0$ and $0$ otherwise, ...
-1
votes
1answer
41 views

Billingsley Exercise 8.8 (Markov Chains)

I am studying from Billingsley and would like some hints on the following exercise. Suppose $S = \{0,1,2,...\}$, $p_{00} = 1,$ and $f_{i0} > 0$ for all $i$. Here, $S$ represents the state ...
3
votes
1answer
67 views

Stock market trading / Casino betting / Multi-player fun competition possible with the following input? [on hold]

I would like to program some kind of online betting system for fun. Just for the fun factor, I would like the Twitch chat to be the random input (seed). As can be seen here, you can see one possible ...
1
vote
1answer
28 views

Simple Question about Monotone Convergence Theorem

Suppose we have a sequence of (discrete) random variables $X_0, X_1, \dotsc$ over $E$ and $A \subseteq E$. Let $Y$ be some other random variable. Moreover, let $Z$ be a random variable with values in ...
0
votes
0answers
39 views

Can you verify the combinatoric recurrence?

There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical. ...
0
votes
0answers
11 views

How are Chi Square probabilities calculated?

What steps would one follow to calculate the values in a Chi Square probability table such as https://people.richland.edu/james/lecture/m170/tbl-chi.html? Say you had 15 degrees of freedom and wanted ...
0
votes
0answers
22 views

Is it possible to exchange a sum in a conditional expectation

Let $X_1, X_2, \ldots \geqslant 0$ and $Y$ be RVs over $\mathbb{R}^n$. Then is it true that $\mathbf{E} \left[ \sum_{i = 1}^{\infty} X_i \mid Y \right] = \sum_{i = 1}^{\infty} \mathbf{E} [X_i \mid ...
0
votes
1answer
21 views

Picking a Winner [on hold]

Two friends are playing a game where they try to predict the score when two-sided dice are rolled and the numbers added together. If one of them guesses the score correctly, they get the point. a) ...
0
votes
2answers
49 views

Prove (or disprove) that $\mathbb{E}[X]\geq 0$ for positive random variable.

Let $X$ be a random variable such that $X\in[0,1]$. I was wondering if $\mathbb{E}[X]$ must be $\geq0$. Since $X$ is a positive random variable, we can apply the Markov-inequality: for each positive ...