Tagged Questions

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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2answers
31 views

Probability (X >Y) when X and Y have the same distribution?

This is a problem from HW4 Joe Blitzstein's Harvard Stat 110 course. Let X be a random day of the week, coded so that Monday is 1, Tuesday is 2, etc. (so X takes values 1, 2, . . . , 7, with equal ...
0
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0answers
10 views

Is there of list of probability distributions ranked by “tightness of fit” for a given SD?

Given the mean and standard deviation of a distribution, we know from Chebyshev's theorem that at least a half of the values will be between between ($\mu - \sqrt{2}\sigma, \mu + \sqrt{2}\sigma$). If ...
2
votes
1answer
17 views

Probability of $B$ winning a series of games

$A$ and $B$ are two players. The probability of $A$ winning a particular game against $B$ is $1/3$ and the probability of $B$ winning the game is $2/3$. They play a series in which the rules are ...
0
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0answers
6 views

Definition and implications of Infimum

Let $\theta$ be a parameter. Let $X_i$ be a random variable. Let $F$ the joint probability distribution of data $\{X_i\}_{i=1}^{n}$. Let $(\theta,F)\in \mathcal{F}$. Let ...
1
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0answers
13 views

How to find the $E(N)$ using $E(M)$ where the $M$ and $N$ follow slightly different scenarios

An author sends his first manuscript to a large number of publishers, $C, D, E, ...$ , in turn, only approaching each one, after the first, if the one before has refused it. There is a constant ...
0
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1answer
16 views

Probability of intersections of independent events with a twist

I'm trying to solve this problem: Let $(A_n)$ be a sequence of independent events. Show that if $I$, $J$, are (finite/countable) disjoint sets, then $$ {\mathbb P}{\Large[}~\bigcap_{i \in I}A_i ...
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0answers
2 views

Distribution under null-hypothesis and type 1 error

Given random variables $X_1,...,X_n \overset{i.i.d.}{\sim} N(\mu, \sigma^2)$ where the variance $\sigma^2$ is known let the null hypothesis be $H_0: \mu = \mu_0$ For the statistic $T=\sum_{i=0}^nX_i$ ...
0
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1answer
15 views

Interpretation of Standard Deviation independent of the distribution?

Is there any intuitive way to interpret the standard deviation regardless of the probability distribution? So for example, for the normal distribution, I know how to interpret being within 1 standard ...
1
vote
1answer
8 views

Discrete Bivariate Distributions, find constant $c$

I am given $f(x,y) = c(x + y)$, and I have to find constant $c$ such that $f(x,y)$ satisfies the conditions of being a joint pmf for two discrete random variables $X$ and $Y$. $x = 1$, 2, 3, and $y ...
0
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0answers
21 views

Find two random variables with same distribution…

I am looking for two random variables and a probability measure $P$, such that $X$ and $Y$ have the same distribution, $X$ and $Y$ are dependent and $X$ and $Y$ are uncorrelated. I tried to take the ...
1
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0answers
22 views

Computing a Finite Expectation

Assume $1\leq\ k<m<n$ are positive integers and $X_1,X_2,...X_n$ are i.i.d. Geometric($p$) random variables. For all $j\geq\ k$ define $I_j=[(i_1,i_2,...,i_k):1\leq\ ...
1
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2answers
19 views

Solving for N in a binomial distribution

Mid-term study... Two dice are rolled. How many times must the dice be rolled so that the probability of getting a sum of 10 or greater on at least one roll is larger than 0.9? So am I correct ...
1
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1answer
24 views

Card Game Probability 13 Card Hand

Me and my friends play a four person poker style card game. Each person is dealt 13 cards, and it is a standard trump card game. Now, as the standard, a five card straight beats a five card flush, but ...
0
votes
1answer
20 views

proof of weak convergence of probablity measure [duplicate]

Let $(\mu_n)_{n\in\Bbb{N}}$ be a sequence of probability measure on $\Bbb{R}$ with characteristic functions $(\phi_n)_{n\in\Bbb{N}}$. Assume that $\lim_{n\rightarrow\infty}\phi_n(t)=1$ for all ...
0
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0answers
8 views

Sum of Lomax random variables

Suppose $X_1,X_2,\cdots X_n$ are $n$ i.i.d Lomax random variables with pdf $f(x)=\frac{m}{(1+x)^{m+1}},x\geq 0,m\in \mathbb N$. I need to determine the pdf (or cdf) of the sum $S_n=\sum_{i=1}^{n}X_i$. ...
1
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0answers
4 views

Conceptual Question on Cramer rao lower bound for performance measure

In system identification, parameter estimation I have found in several papers that an analytical bound is derived which is the CRB of the error variance of the estimates. For, optimal performance of ...
0
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1answer
23 views

Probability from multiple trials

This questions is from a practice mid-term that I don't have a solution to. A monkey in a research lab is given 6 tiles with the letters AAABNN. On each trial the monkey randomly arranges the ...
1
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1answer
13 views

A hand of six cards is dealt from a standard poker deck. Find formula for p_(XYZ) (x,y,z).

A hand of six cards is dealt from a standard poker deck. Let X denote the number of aces, Y the number of kings, and Z the number of queens. a) write a formula for p_(XYZ) (x,y,z). b) Find ...
-3
votes
2answers
17 views

If 4 balls are drawn without replacement, what is the probability that at least 3 black balls are drawn? [on hold]

There are 9 black balls and 10 red balls in an urn. If 4 balls are drawn without replacement, what is the probability that at least 3 black balls are drawn?
0
votes
1answer
25 views

lottery game probability

In the "Make Money Game," the winning number is four digits, each selected at random from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, e.g. 0-3-9-6, 0-0-6-0, 9-4-7-9. A player may place any of the following types ...
0
votes
0answers
19 views

Lack of memory of a geometric distribution, proving a general case.

I have to prove this for a general value so $P(X > j+k | X>j) = P(X > k)$ Using the conditional probability I get that $P(X > j+k | X>j) = \dfrac{P(X > j+k) \wedge P(X > ...
2
votes
1answer
23 views

Frankie and Johnny game. What should Johnny strategy if he wants to minimize his expected loss?

Frankie and Johnny play the following game. Frankie selects a number at random from the interval $[a, b]$. Johnny, not knowing Frankie’s number, is to pick a second number from that same inverval and ...
0
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0answers
10 views

MTG: Probability of drawing a card with enough mana to play it (part 2)

This is a continuation of the original question: Let's assume we have the same scenario 12 White 13 Black 3 Spell Card 32 Other cards Based on the answers in Part 1, I can now answer that ...
4
votes
2answers
78 views

Probability brainteaser

I find this braintease on the internet and do not know how to solve it. For the second question, My first thought is to deduct from the situation when there is only 2 slots, then 3, 4, .., n,.. slots. ...
1
vote
1answer
18 views

What is the probability of success?

If I have 12 Possible questions, of which 5 are asked and I only need to answer 2 of them, what is the probability of my success (i.e., I am able to answer 2 of the 5 asked questions) if I learn 2 of ...
1
vote
1answer
25 views

Deck of Cards Probability Question - Probability of Getting At Least 2 Queens

There was actually another question like this but the final answer a person mentioned was incorrect and I was confused as to how he got it. Can any answers explain how they got there? I'd like to ...
2
votes
3answers
20 views

Symmetric Distribution of Random Variable

Prove: Let $X$ and $Y$ be random variables with the same distribution. If $X$ and $Y$ take only two values​​, then $X - Y$ are symmetrically distributed around zero. Note: 1 - You can use ...
2
votes
2answers
15 views

Probability of an event happening

Studying for a mid-term with a practice test, and there's no solution, so I want to make sure I have this right. A fire alarm has the property that it will ring with 99.5% probability, if there is ...
1
vote
1answer
10 views

Distribution of random variable $Y$ passed throught distributin function of $X$

If \begin{align*} F(x)=P[X \le x] \end{align*} is continuous in $x$, show that $Y=F(X)$ is measurable and that $Y$ has uniform distribution \begin{align*} P[Y \le y]=y, \, 0 \le y \le 1. ...
0
votes
1answer
19 views

Expected value of a sum of random events

Suppose there's a market that has decided to award its most loyal customers. The market sells a certain type of breakfast cereals with a single token in each box. There are n different types of ...
1
vote
3answers
16 views

Standard deviation…

I have this random variable $X = \{-1, 0, 1\}$ with uniform repartition $p(X = -1) = p(X = 0) = p(X = 1) = \frac{1}{3}$. Expected value is $$E[X] = \sum_{i\in\{-1,0,1\}} x_ip_i = 0$$ Then variance ...
0
votes
1answer
12 views

Determining the Likelihoods of Different Game States

Suppose a game is played in which Player 1 must gain two points to win and Player 2 must gain five points to win. Both players start with zero points. In any round, Player 1 has a $1/3$ chance of ...
0
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2answers
20 views

simple probability with marbles - requery

There is a post about how to calculate probability with marbles. I doubt the answer and i am asking for a more detailed explanation if possible. Picking marbles without replacement and without ...
0
votes
1answer
48 views

Probably, expected eatings on a roulette wheel

The probability that a roulette wheel stops on a red number is $\frac{18}{37}$ For each bet on “red” you are returned twice your bet (including your bet) if the wheel stops on a red number, and lose ...
0
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2answers
23 views

Conditional Probability - chance for an event to happen

I am learning probabilities at the moment and I have come across this problem: A person takes four tests in succession. The probability of his passing the first test is p, that of his passing each ...
0
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2answers
21 views

Probability CDF question on highest number of marbles pulled out

I'm kinda stuck on this problem. Here goes: An urn contains n marbles, numbered 1, 2, . . . , n. Suppose k < n marbles are drawn from it at random without replacement. Let X denote the highest ...
0
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2answers
43 views

Probability with expected value for diagnostic tests

Two percent of the population has a certain condition for which there are two diagnostic tests. Test A, which costs $1 per person, gives positive results for 80% of persons with the condition and ...
1
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2answers
33 views

A random variable $X$ uniformly distributed over the interval $[0, 2\pi]$

A random variable $X$ distributed over the interval $[0, 2\pi]$ a) the pdf of $X$ b) the cdf of $X$ c) $P(\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$ d) $P(-\frac{\pi}{6} \leq X \leq \frac{\pi}{2})$ ...
1
vote
1answer
11 views

Probability: How to find what proportion is between the 2 values

Assume that head sizes (circumference) of new recruits in the Canadian armed forces can be approximated by a normal distribution with a mean of 22.8 inches and a standard deviation of 1.1 inches. ...
1
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1answer
21 views

Speed of convergence in probability

Let $X_i$ be a random variable. Let $\{X_i\}_{i=1}^{n}$ be a sample of observations i.i.d. over $i$ with $E(X_i)=\mu$. Let $\bar{X}_n:=\frac{1}{n}\sum_{i=1}^{n}X_i$. Let $\{A_n\}_{n \in ...
0
votes
1answer
16 views

What is the probability of at least 4 events occurring in 6 tries, given that $P(\text{occurring})=0.7$?

Problem: The plant is capable of growing seed 70% of the time. Calculate the probability that out of 6 tries, at least 4 seeds will be grown. A=seed is grown $$P(A)=0.7$$ From where do I start with ...
0
votes
2answers
14 views

Probability of defective of 1 item after picking 5 items out

I had a quiz and one of the question is : If I have 25 items among them 10 are defective. The question is I pick 5 out of 25 and test them, what is the chance of 3 of them are defective. The ...
0
votes
1answer
38 views

Probability - Diagnostic Tests, expected cost per person

Assume that for a randomly selected person: $P (D) = 0.02$, $P (R\mid D) = 1,$ $P (R\mid D') = 0.05$ So that the inexpensive test only gives false positive, and not false negative, results. ...
0
votes
2answers
21 views

Relations among notions of convergence

Let $\{A_n\}_{n \in \mathbb{N}}$ be a sequence of real numbers such that $\lim_{n \rightarrow \infty}A_n=0$. Does this imply that $plim_{n\rightarrow \infty}A_n=0$, where $plim$ is the probability ...
1
vote
2answers
39 views

Expected Value and Variance - Finding expected winnings

A game is played where a fair coin is tossed until the first tail occurs. The probability $x$ tosses will be needed is: $$f(x)=(0.5)^x;x=1,2,3,\ldots$$ You win $2^x$ dollars if $x$ tosses are ...
0
votes
1answer
19 views

What is the probability in the following case?

Given $100$ cells, each cell can contain the values $0$ or $1$ with $0$ - with a probability of $0.96$ $1$ - with a probability of $0.04$ How can I calculate the probability of having at least one ...
0
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0answers
19 views

Poisson Distribution word problems

During rush hour the number of cars passing through a particular intersection23 has a Poisson distribution with an average of 540 per hour. (a) Find the probability there are 11 cars in a 30 second ...
0
votes
1answer
36 views

Application of Slutsky's Theorem

Let $X_i$ be a random variable. Let $\{X_i\}_{i=1}^{n}$ be a sample of observations i.i.d. over $i$ with $ \mathbb{E}(X_i)=\mu$ and $Var(X_i)=\sigma^2>0$. Let ...
0
votes
1answer
23 views

Probability of accessibility

Between A, B, and C, there are the following highways: A – B, A – C, and B – C. During monsoon, when there is heavy rain, each of the road gets blocked independently with probability $p$. What is ...
1
vote
3answers
38 views

Find the pdf of T = X + Y

Let (X,Y) be a random point chosen uniformly on region R = {(x,y) : |x| + |y| <= 1}. I need to find the pdf of T = X + Y. I know the joint density is just equal to 1/(area) = fxy(x,y) = 1/2 for ...