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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1answer
28 views

AMC12B Problem, probability

An unfair coin lands on heads with a probability of $\tfrac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the value ...
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1answer
167 views

3 urns, each with 4 balls. select one ball from each

Three urns are labeled $1,2,3$. Each urn contains $4$ balls labeled $1,2,3,4$. A ball is drawn from each urn such that any ball is equally likely to be drawn. The number on the ball is compared to the ...
2
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2answers
79 views

4 Sided dice - flawed logic?

If we had a 4 sided dice (numbers 1,2,3,4 on the faces) and rolled it 6 times and recorded the results. What would be the probability that we rolled the same number 3 or more times? (Such as rolling ...
3
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1answer
224 views

Probability with flipping a coin 6 times

I have a fair coin and I flip it 6 times, here are the events: $Pr(A)$ = "the coin comes up heads at least 4 times" $Pr(B)$ = "the number of heads is equal to the number of tails" $Pr(C)$ = "there ...
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2answers
79 views

Optimizing Choice of Life Partner

In this link, Hannah Fry mentions that a mathematical argument has been made towards the probabilistically optimal strategy for picking someone to settle down with. The claim is as follows: If we ...
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1answer
69 views

Truncated normal mixture distribution and simulation

Let $X_1, X_2$ and $X_3$ are independent random variables and $X_i\sim N(\mu_i, \sigma_i^2)$. Let $Y$ be a normal mixture random variable that its cummulative distribution function (CDF) is given by $...
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1answer
46 views

Probability formula for this problem

Kindly guide me in solving the following problem. I am learning probability so, i would appreciate a bit of explanation. There are two men M1 and M2. Each man has got two buckets called A and B. ...
2
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0answers
128 views

Large Random Graph is Surely Connected

I'm trying to prove that for a random graph on $n$ vertices with edge-probability $p \in (0, 1)$ is almost surely connected as $n$ grows large. I've tried making an argument using the probability of ...
0
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1answer
18 views

How often I have to do action X, with a $10\%$ chance of a reward, to have a $99\%$ chance of getting said reward?

In a game I play, there is a $10\%$ chance of getting the item I want when I complete a certain action. I'd like to know how often I need to do said action to be $99\%$ sure I will have this item.
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0answers
60 views

What is the probability of a repeat occurring at least once in the decimal expansion of an irrational number?

Let's take pi as an example. Pi = 3.14159... What is the probability that in the decimal expansion of pi, that there exists some number n such that that after the nth digit of the decimal expansion (...
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3answers
70 views

Showing the limit of an event is zero

Let $\{ A_n \}_{n \geq 1 }$ be a sequence of events which are pairwise disjoint and let $P$ be a probability measure. Show carefully that $\lim_{n \to \infty} P(A_n) = 0$. Attempt to the solution: I ...
2
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0answers
71 views

How can I show/solve this equation?

I need help to prove the following equation. $X_n$ is an iid random variable, with: $$\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=\frac{1}{2}$$ Show: $$\mathbb{P}\left(\frac{M_n}{\sqrt{n}}>x\right)=2\...
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2answers
95 views

putting 4 balls into 4 boxes, each ball have a fair probability to fall into each box. Calculate the expected value of empty boxes.

Given the following question: putting 4 balls $\{ball_i\}_{i=1}^4$ into 4 boxes $\{box_k\}_{k=1}^4$. Each ball $ball_i$ have a fair probability to fall into each $box_k$, independently to ...
0
votes
1answer
36 views

In a box there are balls numbered m to n (n>m) and with returns 2013 are drawn and the numbers on them are registered..

What is the probability that the largest and smallest registered numbers are $k\ and\ l (m\leq l <k\leq n)?$ Do the assignment in the case that the balls are drawn without returns, $2012\leq k-l$. ...
0
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1answer
61 views

Proving that the Markov chain is recurrent - Confusion/Help

Giving the following transition matrix [ 0.9 0.1 ] [ 0.8 .2 ] Classify the states From drawing the graph I know that both stats are recurrent. However I'm really failing to prove mathematically ...
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0answers
25 views

Moment determinacy of joint distribution implies moment determinacy of projection

Let $X$ be an $n$-dimensional random vector with a distribution $P$ that is uniquely characterized by its moments. Consider the scalar random variable $\xi_1 X_1 + \dots \xi_n X_n$ with a vector $\xi \...
3
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1answer
247 views

How to prove this Brownian motion convergence?

Let $W_t$ be a Brownian motion. How do I show the following? $$ \alpha > \frac{1}{2} \Rightarrow \lim_{t\rightarrow\infty} \frac{W_t}{t^{\alpha}} = 0 \text{ a.s.} $$ Showing convergence of this ...
0
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1answer
37 views

How many events are associated with a Sample Space of $n$ outcomes?

For example if $n = 2$, the amount of events that may occur is $3$: $S = \{a, b\}$ $E^* = \{a, b, ab\}$ So what are the maximum amount of events that may be derived if the number of samples $= n$?
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4answers
94 views

Probability that exactly k men are chosen by 5 women

Five young women and three young men are friends. One night, each of the women calls one of the men, whom she chooses at random. Find the probability that exactly k men are called for k = 1,2, and 3. ...
0
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0answers
30 views

Central limit theorem with multiple means and variances

How does the Central limit theorem work when there are multiple random variables, each one with a different mean and variance? For example, let $X_1, X_2$ and $X_3$ be random variables that represent ...
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1answer
87 views

The expected value of $1/(1+X)$ where $X$ has geometric distribution

Let $X$ be a random variable with a geometric distribution with parameter $p$. Calculate $\Bbb E[1/(1+X)]$ I've looked at answers online but still do not understand how to solve this. ...
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0answers
47 views

How to find covariance of sample mean and sample standard deviation

I have a question to find the covariance of sample mean and sample standard deviation based on the following: I have tried something on my scratch paper, but for some reason, I cannot upload on here....
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2answers
32 views

Probability: Determine independent event from isosceles triangle problem

A point is selected at random from the triangle {(x,y):0 <=x <= y <=1}. Let E be the event that a selected point has the x coordinate is less than 0.5 and F be the event that y-coordinate is ...
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0answers
27 views

Eigenmatrix of a Wishart Matrix

A lot of papers mention statements of the form "The Eigenmatrix of a Wishart matrix are uniformly distributed on a sphere of $P$ dimensions" but I am having a hard time finding the conditions under ...
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1answer
26 views

Derivated distribution question

I'm having trouble with the following problem: "The salary of a group of employees is represented by the continuous random variable $X$ with probability density function $f_X(x)= \begin{cases}x/2 &...
3
votes
1answer
38 views

R (Stats) - Read anova table and prove Ha

Don't down-vote me for not including more context--You don't need it (unless you're curious for more info). I have a dataset and have the following output in the anova table: (the stats are based on ...
4
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1answer
28 views

Determining te probability that a message can not be corrected

A bit error occurs with probability $10^{-7}$ . A message consists of 8000 bits. Upto three bit errors can be corrected at the receiver with FEC (Forward Error Correction) code in the message. ...
0
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2answers
120 views

Are these two events independent?

Let n ≥ 3 be an integer, consider a uniformly random permutation of the set {1, 2, . . . , n}, and define the events ...
2
votes
1answer
22 views

Ratio between $k$th highest number among $n$ and $n+1$ samples

Let $n\geq k$ be fixed positive integers, and let $X$ be a distribution on $[0,1]$ that is not the constant $0$ distribution. Let $E_n$ denote the expected value of the $k$th highest value among $n$ ...
1
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2answers
124 views

Random variable - what does mean the condition in definition

The definition of random variable in my book is: This is a function $X: \Omega \rightarrow R$, such that $ \{ \omega : X(\omega) < x\} \in S $ where $S$ is sigma-algebra, $\Omega$ is sample space ...
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1answer
73 views

Winning in roulette when betting on one number infinitely

Suppose you join a game of roulette. You choose to only bet on one number all night until you win once, then you stop. Say the probability of winning is 1/37, how will the probability distribution ...
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0answers
56 views

Betting Odds decided by the people who bet - how2compute accuracy?

There is a betting-site which calculates the "odds" for two teams by how much money is set on each team. For example, 100 people place one dollar each, 75 on team A and 25 on team B. Therefore, the ...
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0answers
29 views

How to think about and represent probability of an event happening

Please bear in mind I am neither a statistician nor a mathematician. Plus I thought there was a Stats SE site I could post it to, but I can't find one. Context I have got into a "debate" about how ...
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1answer
465 views

How do I prove that a martingale has a constant expected value?

I can´t prove that a martingale has constant expected value. $$ \mathbf{E}[M_t]=\mathbf{E}[M_0] $$ Thanks people.
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1answer
67 views

Solution for the Tall movie-watchers problem!

The question which I write is a changed format of what I have faced, in my project. I changed it to an attractive question and named it by "tall movie-watchers problem": Suppose that a small cinema ...
1
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1answer
54 views

Conditional probability of a union of two independent events

Suppose $A$ & $B$ are independent events, what is the $P(B \mid A \cup B)$ ? In other words probability of $B$ given $A$ union $B$. My idea was that $$\begin{align*}P(B\mid A \cup B) &= (P(B \...
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1answer
213 views

How to calculate a pair of cards contains at least one ace?

A pair of cards are simultaneously drawn from a deck of 52 cards three times in a row. The drawn cards are returned to the deck. What is the probability that two of three pairs contain an ace? For ...
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2answers
28 views

definition of negative binomial in probability karr

The book defines the probability of the negative binomial as: $$P\{X=k\}={{k-1}\choose{n-1}} p^k (1-p)^{k-n}$$ but where does the ${k-1}\choose{n-1}$ come from? It's quite different to wikipedia's ...
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0answers
43 views

Check if a distribution is discrete or continuous from the characteristic function of the distrution?

Is it possible to check if a distribution is discrete or continuous from the characteristic function/Laplace transform of a distribution?
0
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1answer
27 views

Find $E[W|X>Y]$ where $W = X+Y$ and $X,Y \sim \exp(2)$ independently

I need an idea on how to solve following conditional expectation $E[W|X>Y]$ where $W = X+Y$ and $X,Y \sim \exp(2)$ and $X$ and $Y$ are independent. Thanks.
2
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1answer
43 views

“The three shooters algorithm”

As requested at http://stackoverflow.com/questions/29111313/the-three-shooters-algorithm?noredirect=1#comment46504341_29111313 I have moved that topic into here: Like the title says, you've got 3 '...
0
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1answer
52 views

Dice Probability--Trying to understand the choose function

This is the Question: Three dice are thrown. What is the probability the same number appears on exactly two of the three dice? I know three ways of answering this question (multiplication rule etc..)...
3
votes
1answer
856 views

If P(A) = 0 or 1 then A and B are independent

The question is: Let A,B and C be events in S then prove if P(A) = 0 A and B are independent. Here's my working: For two events to be independent P(A ∩ B) needs to equal P(A)P(B). Since P(A) = 0, P(...
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2answers
71 views

Ratio between highest number among $n$ and $n+1$ samples

When $n$ numbers are drawn independently and uniformly from $[0,1]$, the expected value of the highest number is $A=n/(n+1)$. When $n+1$ numbers are drawn under the same condition, the expected value ...
6
votes
3answers
3k views

Expected value of the number of flips until the first head

Suppose we flip a coin until we see a head. What is the expected value of the number of flips we will take? I am pretty new to expected value, so I tried to evaluate it by multiplying the ...
1
vote
1answer
77 views

Fair coin, three consecutive heads, p(TAILS=1)

Stuck on this question. The experiment of tossing a fair coin until three consecutive heads appear is performed. Let X be the number of tosses, and Y be the number of tails that appear. Find the ...
0
votes
1answer
43 views

finding expectation of a product of random variables

Suppose $X$ is a geometric random variable with probability of succes $p$. I want to show that for $r = 2,3,4,... $, we have $$ \mathbb{E} \{ X(X-1)...(X-r+1) \} = \frac{ r! p^r}{(1-p)^r } $$ Since $...
1
vote
1answer
70 views

Smallest n to align sample mean with population mean

There's a question in my book that I just do not understand. This is it in its entirety: Let $ \bar{X} $ be the sample mean of a random sample of size $ n $ from a normal distribution with a variance ...
0
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1answer
62 views

probabilty mass function of random variables

The random variables X and Y have a joint probability mass function given by the table below: Find the probability mass function of Z = X+Y. What is P(Z=4)? Find E(Z). I've tried to do the ...
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0answers
133 views

Expected number of coin tosses to obtain first tail when probability of heads is $p$

Suppose a coin is tossed independently and repeatedly with probability of heads equal to $p$. What is the probability of only heads in the first $n$ tosses? $Try:$ I think since $ p$ is probability ...