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Probability Question on Stanford Math Tournament

Here is how I would approach it: We have a Markov process. $P_n\pmatrix {\text {smiley}\\\text {no smiley}} =\pmatrix{\frac 23& \frac13\\ \frac 13&\frac 23}P_{n-1}\pmatrix {\text {smiley}\\\...
user317176's user avatar
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Probability Question on Stanford Math Tournament

Alternative approach: There is a smiley face on day 10 if and only if there are an even number of flips in the 9 days between day 1 and day 10. For $~k \in \{0,1,2,3,4\},~$ the probability of having ...
user2661923's user avatar
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1 vote

Probability Question on Stanford Math Tournament

A quick observation of running numbers for $n = 3$ it reveals that The number of times there are flips (by which I mean a smiley face to no smiley face) within these events. Besides the first day, ...
Satish Ramanathan's user avatar
0 votes

Probability of an event in sample space of an experiment that we wait for something particular to occur.

Think of it as an infinitely repeated Bernoulli experiment (you can, and perhaps should, model this entire problem with a geometric distribution and derive all results rigorously). $E_n$ is the event ...
kalkuluss's user avatar
1 vote

Probability of an event in sample space of an experiment that we wait for something particular to occur.

Your solution manual overcomplicates this. I do not have the text but I imagine there is an implied solution process given the section it's in. The event happens or it does not at each "roll"...
David P's user avatar
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1 vote

Commutative diagram involving order statistics

It works because the cdf $F$ is an increasing function, that is: $$X_1 < X_2 \iff F(X_1) < F(X_2)$$
NN2's user avatar
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1 vote

Optimal strategy for uniform distribution probability game

The problem can be formulated as follows: Let $(X_t)_{t \geq 0}$ be a sequence of iid random variables, each is uniformly distributed on $[0,N]$. Adam is the guy who tries to maximize the expected ...
LNT's user avatar
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1 vote

Probability of a random ball being red

We can say an event is probable if it has a probability more than $\frac12$ of happening.'The ball is probably blue or green' doesn't imply that either 'the ball is probably blue' or 'the ball is ...
Zoe Allen's user avatar
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2 votes

Probability of a random ball being red

You asked for a probability of observing a color--this implies an outcome that is a single identifiable color out of the set of all colors represented in the bag. If I ask you, what are the elements ...
heropup's user avatar
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1 vote

Probability of a random ball being red

Let us clearly define: Event $A$ is more likely (more probable) than event $B$ when $P(A)> P(B)$. If event $A$ is more likely than event $B$, then event $A$ is more likely to occur than event $B$....
Amir's user avatar
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0 votes

General rule for finding out a probability distribution

I agree with @Martin Modrák in that the CDF approach is quite general, and with @Masacroso in that the convolution approach may not be of sufficient generality outside the independent world. But in ...
Zack Fisher's user avatar
2 votes

A Normal Distribution of Normal Distributions

If you agree that $\sigma>0$, then $\sigma$ cannot follow $N(\theta_3,\theta_4)$, which is supported on $\mathbb{R}$, rather than on $\mathbb{R}^+$. So the question setting has an issue that ...
Zack Fisher's user avatar
2 votes

Commutative diagram involving order statistics

I don't think there is anything special about random variables or CDFs here. For any list of distinct [non-random] numbers $x_1, \ldots, x_n$ and any increasing function $f$, we have $\text{sort} \...
angryavian's user avatar
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1 vote

The probability of multiplicity of the sum of the dice

It is a Markov chain with 5 states. An "active state" for each remainder $\mod 3$ of the running sum and two ending states indicating whether the final sum is $0 \pmod 3$ or not. I believe ...
ploosu2's user avatar
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0 votes

The probability of multiplicity of the sum of the dice

$\begin{array}{|c|c|c|c|c|c|c|}\hline A|B & 1&2&3&4&5&6 \\ \hline 1&2&0&1&2&0&1\\ \hline 2&0&1&2&0&1&2 \\ \hline 3&1&2&...
true blue anil's user avatar
2 votes
Accepted

When is $\mathbb E[F(S)\mid S=s]= \mathbb E[F(s)]$ true?

I am looking for a sufficient condition under which it is true that $$\Bbb E[F(S)\mid S=s] = \Bbb E[F(s)]$$ $S$ can be seen as a "selector" r.v., each value of which just picks out one r.v. ...
r.e.s.'s user avatar
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0 votes

Optimal strategy for uniform distribution probability game

Let $X$ be a random variable representing Adam's final profit and $y = \mathbb{E}[X]$ denote the expected profit from playing this game. Assume, at a given role, Adam has $v$ dollars. Then, if he re-...
mathz2003's user avatar
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3 votes

Law of Large Numbers for Changing Distributions

Random remarks: Statistics questions probably get better answers on https://stats.stackexchange.com In general, you can put probability distributions on your parameters: that's called Bayesian ...
Guillaume Dehaene's user avatar
3 votes

If You Go Fishing Everyday - What Is The Probability You Know X% Of The Pond?

I am procrastinating from my PhD so I'm typing my comment out. I think a Wright Fisher type model might be a better ansatz for the fish population. (This is the most basic model in population genetics,...
David's user avatar
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1 vote

If You Go Fishing Everyday - What Is The Probability You Know X% Of The Pond?

I can propose the following model. But I allowed myself to modify a bit your conditions in order both to simplify the model and to fix the following model problems: The fishes are integer beings, ...
Alex Ravsky's user avatar
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0 votes

Confidence Interval - Misunderstandings

There is a proof by contradiction to establish that "A 95% CI does not mean that the CI itself has a 95% probability to capture the population parameter". Let us start with the assumption ...
Surm's user avatar
  • 123
2 votes

How long will it take for a coin to repeat a certain behavior?

This is not exactly in line with your approach (Markov chains and mixing times), but it might be useful to gain some intuition. First, notice that your initial question ("How long will it take ...
leonbloy's user avatar
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3 votes

Probability of 3 darts landing in the same half of the board

The “darts on a board” version is equivalent to the “points on a circle” version. When you throw darts on a board, you can radially project the darts onto the circumference of the dartboard; these ...
Mike Earnest's user avatar
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4 votes

How long will it take for a coin to repeat a certain behavior?

I think you did this analysis correctly, some thoughts: The study of mixing times is very interesting :) if you want are looking for a rigorous source, have a look at this book: Markov Chains and ...
David's user avatar
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4 votes
Accepted

Understanding implication of the convexity of a class of distributions

Let's, for the sake of simplicity, assume that both ${X}$, and $Y$ are discrete. Then we can write, by the law of total probability, $$ \mathbb{P}(Y=y) = \sum_x\mathbb{P}(Y=y\vert X = x)\mathbb{P}(X=x)...
MrTheOwl's user avatar
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8 votes
Accepted

"Peeling Technique" in Probability

For the first inequality, note that, using standard prob. theory notation, we have \begin{align} \left\{ \exists t\geq 1:\: S_t + \sqrt{4t\cdot\log^+\left(\frac{1}{t\delta}\right)} + t\Delta\leq 0\...
Joseph Expo's user avatar
2 votes
Accepted

Is such a property for conditional expectation true?

Suppose $X$ and $Y$ are $\mathbb{R}$-valued random variables such that $X \in L^1(\mathbb{P})$. Then $$ \mu(A) = \mathbb{P}(Y \in A) \qquad\text{and}\qquad \nu(A) = \mathbb{E}[X\mathbf{1}_{\{Y \in A\}}...
Sangchul Lee's user avatar
8 votes

How to Define Higher-Order Terms Analogous to Expectation and Variance in Probability Theory?

The higher-order generalizations of the expectation and variance are called the cumulants, $\kappa_n(X)$. They can be defined using the logarithm of the moment generating function: $$K_X(t) = \log M_X(...
Qiaochu Yuan's user avatar
1 vote
Accepted

An inequality for a bisected "shifted quadrant" under a continuous symmetric bivariate distribution?

It is sort of sufficient. Edited details: Let $f(x,y)$ be the density and $C_n = \{(x,y): x^2+y^2 \le n\}$. Define $$I_n := \int_{A\cap C_n} f(x,y) \mathrm{dx} \mathrm{dy}~,\quad J_n := \int_{B\cap ...
Sounak's user avatar
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0 votes

Change of Variables and Expectation of Random Variable

So we have the transformation $y=Ax+B$, the relation between probability density functions $f_{Y}(y)=\frac{1}{A}f_{X}(x)$, and the relation between variation $dy=A\phantom{.}dx$. Therefore the ...
acat3's user avatar
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0 votes

Infinite Summation of Almost Sure Convergent RVs

Note that by setting $X_{n,i} = 0$ for $i>b_n$ the statement is equivalent to $\sum_{i=1}^{\infty} X_{n,i} \to \sum_{i=1}^\infty X_i$ almost-surely as $n\to \infty$. In turn, by considering the (...
raj's user avatar
  • 321
0 votes

EM algorithm for estimating worker ability

As for any EM-problem, you should first construct the likelyhood of the data, regardless of whether it's classic or Bayesian inference. So let’s first proceed to compute the likelihood of the problem ...
Egor Larionov's user avatar
4 votes
Accepted

2D Modified Random Walk

When you say "problem", I think that you want to express the probabilities for the moving point to be in such or such position after $n$ steps. The classical technique for these kind of ...
Jean Marie's user avatar
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0 votes

Describing Frequentist vs Bayesian probability using casual dialogue

The initial section before the "gold watch" example seemed to concentrate on parameter estimation, rather than testing. For the watch example, the Bayesian seemed arguing that for "this ...
Zack Fisher's user avatar
5 votes

This expected value has a minimum!

Another way to do this arises from stochastic dominance. Given a positive random variable $X$ whose p.d.f is bounded above by $1$, create a uniform random variable $U[0,1]$ on the same probability ...
Sarvesh Ravichandran Iyer's user avatar
8 votes

This expected value has a minimum!

Here is a variant of @ConnFus's solution. Let $f(x)$ denote the density function of $X$, and let $g(x) = \mathbf{1}_{[0, 1]}(x)$. Then we note: $\int_{0}^{\infty} [f(x) - g(x)] \, \mathrm{d}x = 0$, $...
Sangchul Lee's user avatar
10 votes

This expected value has a minimum!

Because $X$ is positive, $$ E[X] = \int_0^\infty a(x) \,dx \ge \int_0^1 a(x) \,dx $$ where $a(x) = 1-F(x) = 1 - \int_0^x f(u) \,du$, with $a(0)=1$. Now, $f(x) \le 1 \implies a'(x) \ge -1 \implies a(x)...
leonbloy's user avatar
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0 votes

What are the restrictions on covariance matrices of nonnegative random variables?

This can be understood as a special case of the multivariate form of the Stieltjes moment problem. I was able to find Theorem 2.9. of https://arxiv.org/pdf/math/9905215 characterizes the sequences of ...
Martin Modrák's user avatar
0 votes

Confused about a counting problem

We can answer the question by breaking it into a series of smaller questions and combining. Your misunderstanding begins in question 4 below. 1: How many ways are there to combine $40$ players into ...
fleablood's user avatar
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0 votes

Confused about a counting problem

The whole point is that we are just counting the number of possible pairs without bothering about the pairs' position in the line or the order within a pair, that is how we get $\dfrac{40!}{2^{20}\...
true blue anil's user avatar
1 vote
Accepted

Confused about a counting problem

The $10!$ comes up when trying to figure out how many pairings there are when only offense-offense and defense-defense pairings are allowed. Then there are still $20$ pairings, but they partition into ...
Ingix's user avatar
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1 vote

Distance between two vertices picked at random from random graph.

This depends on your random graph model. Peeyush's answer is true for fixed $p$, however in a random graph model, we often take $p \in \Theta(\frac{1}{n})$, then by the same reasoning, sending $n\to \...
Elijah Blum's user avatar
0 votes

Probability apple is delicious given a red apple is produced in a green apple orchard?

Using conditional probability: Given two events $~E_1~$ and $~E_2,~$ you have that $$p(E_1) \times p(E_2 ~| ~E_1) = p(E_1,E_2) \implies p(E_2 ~| ~E_1) = \frac{p(E_1,E_2)}{p(E_1)}. \tag1 $$ Let $E_1~$...
user2661923's user avatar
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1 vote
Accepted

What is the chance of two events happening after X attemps?

They are disjoint (you can only get one, with probabilities $0.05$ and $0.1$), so the chance of neither of them happening is $1 - 0.05 - 0.1 = 0.85$. The probability of neither of them happening ...
user3257842's user avatar
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1 vote

What is the chance of two events happening after X attemps?

Let $a=0.05$ and $b=0.1$ denote the probability of each of the two events. The likelihood that after $n$ trials, the first event has not occurred is $(1-a)^n$. The likelihood that the second event has ...
Einar Rødland's user avatar
0 votes

Is such a property for conditional expectation true?

The equality is valid by the following: $$\lim\limits_{\epsilon \to 0^+}E[X\mid |Y-a|<\epsilon]=\lim\limits_{\epsilon \to 0^+}E[E[X\mid Y] \mid |Y-a|<\epsilon]=E[X\mid Y=a]$$
Speltzu's user avatar
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1 vote

How to Determine the Bounds for Integrals with Multiple Inequalities in Order Statistics Problems?

You have various constraints. Treating $x_3$ as fixed for a moment, the area you are integrating over for $x_1$ and $x_2$ looks like Since you seem to want to integrate with respect to $x_1$ first, ...
Henry's user avatar
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1 vote
Accepted

Understanding wrong approach to "Probability of forming a triangle".

Following your approach, $x$ is uniformly distributed on $[0,L]$ and $y$ is uniformly distributed on $[0,L-x]$. Note this is different to the approach of having $x$ and $y$ both distributed on $[0,L]$ ...
Adam Dougall's user avatar
0 votes

Possibility that all lights $\mathbf{X}=(X_1,X_2,\cdots)$ turn off again with every time turn a light with its number $n\sim\text{geom}(\frac{1}{2})$.

As noted in many other answers, it suffices to show a $\Omega(1/t)$ bound on the chance the process has returned to the starting state at time $t$. This answer gives another way of showing such a ...
Ziv's user avatar
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0 votes

What is the uses of Expected Value in this context?

There are several random variables here: $X$ is a random variable with uniform distribution on the sample space $[-1,1]^p$, the $p$-dimensional cube having edges of length $2$ which are parallel to ...
Eric Towers's user avatar
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