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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1
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0answers
8 views

Is my method of working fine?

Suppose a point $X$ is selected at random from a line segment $AB$ of length $l$ and midpoint $O$. Find the probability that $AX,BX$ and $AO$ form a triangle. My method and working is: Case ...
0
votes
0answers
3 views

Using Conditional Jensen inequality proof the following

$X_1,X_2,\ldots,X_n$ are i.i.d. random variables, $X_1>0$, $E[X_1]=\mu$, $E[X_1^k]<\infty$ for $1<k \leq2$. Proof: $$ E\left[\left(\frac{1}{n}\sum_{i=1}^nX_i\right)^k\right]\leq ...
1
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0answers
27 views
+50

Expected time until beating an initial try

Consider the following problem: Let $X,X_1,X_2,...$ be i.i.d. random variables. We execute the following experiment. One samples $X$. Then, one samples $X_1$,$X_2$ and so on until the first time the ...
0
votes
2answers
13 views

Rolling dice probability by solving inequlity

I was trying to solve a problem where I have to find the probability of the sum of 3 rolls of a die being less than or equal to 9. In order to solve the problem I try first to find the number of ...
0
votes
0answers
8 views

Probability of a sequence of urn draws having some pair of draws with a minium number of “matches”?

I have $U$ urns. Each urn contains some sequentially numbered balls (not necessarily the same count between urns) $1, 2, 3,... N_u$. I draw one ball from each urn $1, 2, 3,...U$ in turn, and note ...
19
votes
3answers
5k views

Probability that a stick randomly broken in two places can form a triangle

Randomly break a stick (or a piece of dry spaghetti, etc.) in two places, forming three pieces. The probability that these three pieces can form a triangle is $\frac14$ (coordinatize the stick form ...
-2
votes
0answers
13 views

conditional probability proof 3 varables

Suppose that a,b and c are dependent variables. P(a|b)=sum(P(a|b,c)+P(c|b)) can anyone explain it how we get it?
1
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1answer
15 views

Covariance of $2$ variables

I am given two random variables $X$ and $Y$. I am also given that $\mathbb{E}(Y|X)=\mathbb{E}(Y)=\mu_y$ and $\mathbb{E}(X)=\mu_x$. So if I need to calculate the covariance of $X$ and $Y$, ...
0
votes
0answers
17 views

How to use joint probability density to check for independent events?

Suppose that the joint PDF of $X$ and $Y$ is as follows: $$ f(x) = \begin{cases} 24xy & \text {$x \geq 0, y \geq 0, x+y \leq 1$}\\ 0 & \text {otherwise ...
5
votes
3answers
501 views

Probability of some die face being missed N or more times in a row in M rolls? [Clarified]

2015/01/28 Clarification: Question rewritten to remove ambiguity that elicited (interesting) responses to a different problem (use edit should you wish to view). From Markus' feedback on the earlier ...
1
vote
2answers
336 views

Finding expected number of white balls drawn before any red ball

A box contains 12 balls: 6 white, 4 black, and 2 red. Draws are made without replacement. Find the expected number of white balls drawn before any red ball is drawn. Similarly, how do I find the ...
4
votes
1answer
29 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...
1
vote
1answer
15 views

Question about asymmetry of chi-square distribution

Let $X_1,\dots,X_n$ be a set of i.i.d. chi-square random variables with $k$ degrees of freedom. Consider the statistic $\arg\max_i\{|X_i/k - 1|\}=X_{\alpha}$. I wonder about the probability that ...
1
vote
1answer
127 views

Markov's Inequality for Negative Binomial distribution

Given that $Y$ follows Negative Binomial distribution (counts y successes before $k$th failure), using Markov's inequality show that for any $q \in[p,1]$, there exists constant $C$, such that ...
-1
votes
0answers
23 views

How to prove stochastic dominance? [on hold]

Consider the set of constant vectors $p_i$ and $\tilde{p}_i$, such that $p_i \succeq \tilde{p}_i \succ \mathbb{0}\; \forall i$ (component wise inequality) and define: $M \triangleq ...
0
votes
0answers
28 views

Conditional Probability Proof for three events

For any 3 events $X, Y$ and $Z$ where $\Pr Z) > 0$, it is required to prove that $$\Pr ((X \cup Y) \mid Z ) = \Pr(X\mid Z) +\Pr(Y\mid Z) - \Pr ((X \cap Y) \mid Z)$$ I am not able to prove ...
-2
votes
0answers
18 views

Martingale Ideas in Elementary Probability [on hold]

There is a non-symmetric version of the probability model in which the probability of success on each trial is "$p$" and the probability of failure on each trial is "$q$" and $p+q=1$. The probability ...
2
votes
3answers
32 views

Probability of balls in boxes

If $12$ balls are thrown at random into $20$ boxes, what is the probability that no box will receive more than $1$ ball? So my book says the answer is: $\displaystyle \frac{20!}{8!20^{12}}$ However ...
21
votes
15answers
3k views

Is the Law of Large Numbers empirically proven?

Does this reflect the real world and what is the empirical evidence behind this? Layman here so please avoid abstract math in your response. The Law of Large Numbers states that the average of the ...
1
vote
1answer
411 views

Long run probability of going to a state from another

We consider the following transition matrix for a markov chain with state space {A,B,C,D,E} : $P= \left( \begin{array}{ccccc} \frac{1}{2} & 0 & 0 &0 &\frac{1}{2} \\ 0 & ...
1
vote
1answer
28 views

Calculate the mean, the median and the quartiles.

Let $D=\{(x,y):x>0,x^2+y^2<1\}$ and let $(X,Y)$ be the random variable with the density: $$f(x,y)=\frac{2}{\pi}1_{D}(x,y).$$ Let $Z=\frac{Y}{X}$. Calculate the mean, the median and the first and ...
2
votes
2answers
24 views

Convergence in law of sample means of random variable

Let $\{X_n | n \in \mathbb{N} \}$ be a sequence of independent identically distributed random variables with density function: $$f_X(x) = e^{\theta - x}I_{(\theta, \infty)}(x)$$ with $\theta > ...
0
votes
2answers
15 views

Multinomial Coefficients Dice Problem

If 7 balanced dice, are rolled, what is the probability that each of the 6 different numbers will appear at least once? My attempt: $p=\frac{7!}{2!6^6}$ So if 6 different numbers need to appear, ...
1
vote
1answer
24 views

Probability and Statistics (Normal Distribution)

Having trouble with the last part of this question. Not sure how the man would divide his pile of vouchers? It seems that you could interpret this question in a lot of ways. Any tips would be ...
2
votes
1answer
38 views

Probability of winning a tie-break in tennis?

The winner of a tennis tie break is the first to get to 7 points and lead by 2. Let $p$ be the probability of player 1 winning when serving, and let $m$ be the probabiliity of player 1 winning when ...
0
votes
0answers
217 views

Expected number of steps for reaching $K$ in a random walk

Assuming steps are $+1/-1$ with a $50/50$ probability. What is the expected step count for reaching $10, 100$ or $K$?
0
votes
1answer
25 views

Calculate the probability given by three random variables

Let $X_1,X_2,X_3$ be IID random variables, each with the density $$f(x)=x e^{-x}\cdot 1_{(0,\infty)}(x).$$ Calculate $P(X_1+X_2+X_3>4,X_1+X_2<4)$.
0
votes
0answers
7 views

Obtaining the Log-logistic distribution from a truncated logistic distribution

Let $$f(x) = \frac{e^x}{(1+e^x)^2}~,~ -\infty \lt x \lt \infty~~~~~(1)$$ be the standard logistic pdf of a random variable $X$. Then one can obtain the pdf of the log-logistic distribution via the ...
1
vote
1answer
43 views

Probability space defined by function from $X$ to $[0,1]$

Let $X$ be a non-empty countable set. If there is a function $f:X\rightarrow [0,1]$ such that $p(S)=\sum_{x\in S} f(x)$ for all $S \in 2^X$, then prove that $(X,2^X, p)$ is a probability space. My ...
10
votes
7answers
63k views

Probability of 3 Heads in 10 Coin Flips

What's the probability of getting 3 heads and 7 tails is one flips a fair coin 10 times. I just can't figure out how to model this correctly.
0
votes
0answers
15 views

Show that for any geometric random variable $X$ and parameter $p, \mathrm{Pr}(X < t) = 1 − p^t$. [on hold]

How to prove the above stated equation? I tried the following : Pr⁡(X(i=1)^(t-1)▒〖Pr⁡(X=i)〗 =∑(i=1)^(t-1)▒〖p(1-p)i-1〗 =1-(1-p)t-1
1
vote
2answers
54 views

Probability of $k$ collisions

Say we have $m$ buckets. We select a random bucket and put a ball in it, we repeat this $n$ times. In the end what is the probability of having at least one bucket with exactly $k$ balls? I have ...
0
votes
0answers
9 views

Question on Standard Brownian Motion [on hold]

What is the following probability $P [ W(2) >0 \ \text{and}\ \ W(1) <0]$?
2
votes
1answer
23 views

Can two nodes in a Markov chain have transitions that don't total 1?

In all the Markov diagrams I see, the transitions from state A to B always total to one. Just one of many examples, this image ...
5
votes
1answer
93 views

Showing a Borel-Cantelli-esque result for not necessarily independent random variables

$A_1, A_2,\dots$ are events that are not necessarily independent. Prove that $$P(A_n \text{ i.o}) = 1 \iff \sum_n P(A_n \cap B) = \infty \text{ for all $B$ with $P(B)>0$}.$$ $(\Rightarrow)$ ...
0
votes
1answer
31 views

Estimator for second moment for Poisson random variable

Let $X \sim Poiss(\lambda)$. As, $\displaystyle \sum_{i=1}^{N} X_i $ is sufficient statistic for both mean (and variance) of $Y$, so we can define the unbiased estimate for mean as , $ s=\frac{1}{N} ...
-4
votes
0answers
33 views

Probability of an unbalanced coin [on hold]

Let's say i find a coin on the ground and i flip it 100 times getting 99 heads, what is the probability that the coin is unbalanced?(in particular that the probability of getting head is higher than ...
0
votes
1answer
41 views

How to prove the folowing theorem in probablity? [duplicate]

Show that for any continuous random variable $X$ that takes only positive real values $\int_{0}^{\infty}\text{Pr}(X\geq x)dx=\mu$ where $\mu$ is the mean.
-1
votes
1answer
35 views

Proving $P(A_i)=P(B_i)$ [on hold]

Suppose that $P$ is a probability on a field $F$. Consider three events $A_1,A_2,A_3 \in F$ so that $P(A_i \cap A_j) = 0$ for all $i \neq j$. Let $B_1 = A_1, B_2 = A_2 \cap A^c_1$ and $B_3 = A_3 \cap ...
0
votes
0answers
27 views

How to demonstrate the pdf of $P_{\sigma} (t)=\lambda_c e^{- \lambda_c t} / (1 - e^{- \lambda_c T})$

In $t_c$, there are $n$ expirations of $T$ and the remnant $\sigma$ seen from the above figure. Let the time $t_c$ forms the exponential distribution with parameter $\lambda_c$. How to demonstrate ...
0
votes
1answer
22 views

Doob-Kolmogorov Inequality

Denote by $(X(t),t\ge 0)$ a standard Brownian motion, i.e random variables with the following properties: $X(0)=0$. With probability 1, the function $t\mapsto X(t)$ is continuous on $[0,\infty)$. ...
1
vote
1answer
24 views

Obtain the MGF of $Y$.

Let $X$ be a random variable whose probability density function is: $f_{X}(x)=e^{-x}$ Then, obtain the Moment Generating of function of $Y=1-e^{-X}$ What I did: We can find a bound for $y$ using ...
0
votes
0answers
21 views

Limit definition of Sets.

Proposition 1.32 $X_{n}\xrightarrow{a.s.} X$ if and only if for any $\epsilon>0$ $P( | X_{n}- X |<\epsilon, \; \forall n\geq m )\rightarrow1$ $as$ $ m\rightarrow\infty$ Proof. Suppose first ...
1
vote
1answer
22 views

Sum of uniform random variables $U(0,1)$ and $U(0, a)$

The problem I have is: $X \sim U(0,1), Y \sim U(0,a)$ are independent random variables. Find the pdf of $X + Y$. I've got stuck in an integral-problem, and will show you what I've tried. Skip to the ...
0
votes
1answer
24 views

Sigma-fields and probability

I'm unsure what this question asks of me. For (i) I have given a power set with 16 elements in terms of a,b,c and d. I don't understand what I need to do for (ii). I believe (iii) is fairly ...
5
votes
1answer
52 views

Sum of squares of Binom(n,p) values

Let $x_{n,p}(j)$ be the probability that a random variable distributed according to a binomial distribution with parameters $n \in \mathbf{N}_+$ and $p \in (0,1)$ takes the value $j \in ...
2
votes
1answer
29 views

Probability of winning a game in tennis?

Suppose there is a tennis singles match, where Player A plays a single game against Player B. The probability that player A will win a single point is $x$, and thus $1-x$ is the probability that ...
1
vote
1answer
23 views

If the probability of 3 events with non-zero probability equals the product of the individual probabilities, are they also pairwise independent?

Consider three events $A$, $B$, and $C$, none of which has a zero probability. If $A$, $B$, and $C$ satisfy $\Pr(A \cap B \cap C) = \Pr(A) \cdot \Pr(B) \cdot \Pr(C)$, does this imply that the three ...
0
votes
1answer
21 views

ways to choose 4 people including two or more male from N people with 1/3 male 2/3 female

A sample of four people is randomly drawn from a population of N > 4 people.Assume that 1/3 of the total population is male, and 2/3 is female. (To simplify things, let’s assume that N is always ...
0
votes
1answer
6 views

CDF of the kinetic energy of a particle under uniform distribution

We are given that X~Uniform[2,3] and the kinetic energy is $T=\frac{1}{2X^2}$ I tried the following: $P(T\leq a) = P(\frac{1}{2X^2}\leq a) = P(-\sqrt{2a}\leq X \leq \sqrt{2a}) = ...