New answers tagged probability
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Monty Hall Problem - $4$ doors where you pick one by one
The best strategy is to stay before the last choice and then switch, because you're reducing the problem to that of $3$ doors, but since this is a variant with $4$, the probability of winning by ...
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Hazard ratio as the odds of dying first
Take an example of two events whose time of occurrence are independently exponentially distributed with parameters $\lambda_1$ and $\lambda_2$ so $F_1(t)=1-e^{-\lambda_1 t}$ and $F_2(t)=1-e^{-\...
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Is the variance of a product of a normal and an arbitrary density uniformly bounded with respect to the mean of the normal density?
I assume that $\sigma=1$ for simplicity, and I denote $f_1(x)=e^{-x^2/2}f(x)dx$ then the new probability $h_{\mu} $ belongs to the exponential family generated by $f_1$. The Laplace transform $L$ of $...
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Accepted
Distribution of N Car Platoons
The probability that at least $n$ cars follow a given one is the probability that the given car's speed is less than all previous $n$ cars. For iid speeds following a continuous distribution, this is ...
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Failure theory in graphs and network
First, let's look at whether $2$ and $4$ are connected to $1$ when looking only at the left triangle (i.e. without considering paths like $1-2-3-5-4$). The possibilities here are:
$C_2$ (only $2$ is ...
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independence of norms of normally distributed vectors
No.
The simplest counterexample is probably to have $X$ and $Y$ both standard multivariate normal $\mathcal N(\bf{0},\bf{I}_n)$ where their magnitudes are independent (each $\sim \chi_n$) but their ...
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Estimate for Overshoot
This follows from Lorden's inequality (Wikipedia), which bounds the overshoot by
$$
\mathbf{E}\Bigl[\sum_{i = 1}^N X_i \Bigr] - B \leq \frac{\mathbf{E}[X^2]}{\mathbf{E}[X]}.
$$
See Chang (1994) ...
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Failure theory in graphs and network
The framework/theory which might be relevant to what you are asking about is Percolation theory. But this seems like severe overkill to your question.
It seems to me that you are asking when the links ...
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What is the flaw in the first solution given below?
This might be a more intuitive way to look at it.
All in all, there are 4 distinct possibilities: A and B are both available, only A is available, only B is available, neither of them are available. ...
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What is the flaw in the first solution given below?
The rule of addition - $P(X\text{ or } Y)=P(X)+P(Y)$ only applies if the events are disjoint, that is, they can't both happen. Here clearly the paths can both be available.
In general, the rule ...
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Why is this not a valid Variance Covariance matrix, and inherently not positive semi-definite?
There have been two good answers on the impossibility of the covariance matrix. On the other hand, OP asked whether there is something "inherently wrong" with the matrix. This suggests that ...
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Why is this not a valid Variance Covariance matrix, and inherently not positive semi-definite?
Suppose we have a set of real r.v.s $\{X_k\}_{k=1}^n$ such that $\text{E}[X_k^2]=1$ for all $k$ and $\text{E}[X_jX_k]=-1/2$ for all $j\neq k$. Then the variance of $X:=\sum_{k=1}^n X_k$ would be
\...
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Need an elegant solution for a geometric probability problem
I think you can simplify somewhat by thinking of a double integral. First define a function: for each pair of points $U, V$ on the square, compute the distance from the segment UV to the circle center;...
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Two player coin flipping game
Let us start by calculating the expected value when you go first, we will call this $E$. We know that there is a 1/2 chance player one wins on the first round. We also know that if player 1 doesn't ...
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Two player coin flipping game
Here's my attempt at a solution.
Let the sample space be sequences of H and T till the first head. $S = \{H, TH, TTH, TT...TH\}$. Now consider a random variable X that calculates the amount earned/...
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Probability of selecting a desired chip
$P(B$ selects $12)$ = $\frac{{1 \choose 1} \times {13 \choose 4}}{{14 \choose 5}} \times \frac{2}{7} = \frac{5}{49}$
As other responses have indicated, a symmetry argument leads directly to the $~\...
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Balls are different or same?
Your answer to the original question is correct, though it might be clearer if written as $\frac69\frac38\frac57\frac26+\frac39\frac68\frac27\frac56=\frac5{42}$ to match your "RWRW or WRWR". ...
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Normal r.v. convergence in law to a normal r.v. implies convergence of mean and variance
Let $N$ be a standard normal rv. We have
$$\Bbb P\!\left(N>\frac{t-\mu_n}{\sigma_n}\right)=\Bbb P(X_n>t)\xrightarrow[n\to\infty]{}\Bbb P(X_0>t)$$
and therefore
$$\limsup_{t\to\infty}\:\...
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Normal r.v. convergence in law to a normal r.v. implies convergence of mean and variance
Because the standard normal cdf is one-to-one, if $X_n$ converges in law to $X$ then $(x-\mu_n)/\sigma_n\to(x-\mu)/\sigma$ for all real $x$. That's a lot of information to work with!
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Finding a CI for variance using Chi squared, when mean is known and samples are normal but not canonized
Note that (A) $\left(\frac{1}{n}\sum\limits_{i=1}^n X_i^2\right) -\mu^2$ is almost certainly not equal to $\frac{1}{n}\sum\limits_{i=1}^n \left(\left(X_i -\mu\right)^2\right)$ even if $\frac{1}{n}\sum\...
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Classical probability
As an extended comment to demonstrate that lulu's answer is better than the solution you saw, here is a simulation in R:
...
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Classical probability
As @lonza leggiera has remarked in a comment, the book answer has computed favorable ways and total ways using stars and bars, which is inappropriate as stars and bars does not yield equiprobable ...
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Classical probability
Alternative approach:
This approach uses Stars and Bars theory, in a somewhat convoluted manner, to attack the problem. On the one hand, I consider the Stars and Bars approach to be significantly ...
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Accepted
Drunk Passenger Follow-up
At each random selection only 3 things of significance can happen:
The drunk person or a person with their seat taken randomly picks seat 1.
The drunk person or a person with their seat taken ...
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Binomial distribution and bernoulli
What is it that you are struggling to prove?
(1) that if a probability space has a random variable $X\sim$Bin($n,p$), then it also has $n$ independent random variables each with Ber($p$) distribution. ...
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How do you show something does or does not hold in finite samples?
They already give a proof in the text, but I think the most concrete (IMO, of course) argument is sort of shoved into a footnote. Perhaps developing it fully would help.
Take $S_1, \cdots, S_n, S_{n+1}...
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Classical probability
To do it your way:
I am, of course, assuming that each letter is equally likely to appear.
First, we ignore the restriction that $J$ should only get one free one.
Consider the entire sample space, of $...
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How to maximize Playerpower while minimizing Crest expenses World of Warcraft
You can formulate this process as a Markov Decision Process. In such a process, you have a state space $S$, an action space $A$, a reward function $R:S\times S\times A\to\mathbb{R}$ and a collection ...
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Problem with 3 urns and balls of 3 different colors
The analysis in the posted question is valid, precisely because the urns, being numbered, are to be considered distinguishable, rather than indistinguishable.
That is, with the urns distinguishable, ...
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Problem with 3 urns and balls of 3 different colors
This depends upon some assumptions made:
If the model is that a child is given these balls, and asked to make all possible numbered heaps of the colored balls, your answer is correct
If, on the ...
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Source for a dialogue of Galileo about distinguishable dice?
The essay in question is titled "Sopra le scoperte dei dadi" ("On the Investigations of Dice"), written prabably between 1613 and 1623, in response to a request for an explanation ...
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Stein's method for non-unit variance
Strictly speaking, Stein's method does not establish convergence, but only provides a bound on the difference between $W_n = \frac{1}{\sqrt{n}} \sum_{i=1}^n X_i$ and $X \sim N(0,1)$ in the Wasserstein ...
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Why am I encountering two different formulas for unordered sampling with replacement?
$\binom{n+r-1}{r}$ is correct, here is a classical explanation. Suppose we are unorderedly choosing $r$ out of $n$ distinct items with replacement, then each possibility is one-to-one corresponding to ...
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When can you add probabilities?
It will be simpler to understand if you take smaller figures, say P(crash) $= \frac1{10}$ and # of flights $=10$
P(it doesn't crash in $10$ flights) $= \left(\frac9{10}\right)^{10} \approx 31.87\%$
by ...
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When is $X_n \sim N(\mu_n, \sigma_n^2)$ uniformly integrable?
Here's the proof of @geetha290krm's comment:
The collection $\{X_n: n \ge 1\}$ where $X_n \sim \mathcal{N}(\mu_n, \sigma_n^2)$ is UI iff $(\mu_n), (\sigma_n)$ are bounded sequences.
"$\...
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What will be the conditions under which- "If A and B are events such that P(A|B) = P(B|A)"
Assume $\mathbb{P}(A) > 0, \mathbb{P}(B) > 0$. The condition for the equality of conditional probabilities can be expressed as:
$$
\mathbb{P}(A \mid B) = \mathbb{P}(B \mid A),
$$
where
$$
\...
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Proof of the delta method
Excellent point! Why is such a $\overline{\theta}$ even measurable? I congratulate you on your rigor. Here is a proof which avoids these difficulties:
Lemma:
\begin{align*}
&\text{i)}\hspace{...
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Money division based on partial game results
One can encode each match as a series of 0's and 1's, where a 0 indicates that S won the game and a 1 indicates that V won the game. This code has between 5 and 9 binary digits. There are 252 matches ...
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Number of tries in key guessing game
Perhaps the simplest way to see that your answer is correct is as follows.
Suppose you choose a uniformly random order to look for the tiles in, and following that order you would take $R$ tries to ...
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Accepted
Money division based on partial game results
This is a very interesting problem, and overnight I have come to a different way of thinking
We have to realise that it is a new game with new rules, starting from $Varun /Sandeep $ having won $1/2$ ...
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Number of tries in key guessing game
I guess this has been sort of highlighted but not stated explicitly. Without any information about where the key is, the order of your guesses and the formation of the tiles in the grid does not ...
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What will be the conditions under which- "If A and B are events such that P(A|B) = P(B|A)"
The textbook answer is wrong.
Assuming $P[A]$ and $P[B]$ are both nonzero so that the conditional probabilities exist, the condition $P[A|B] = P[B|A]$ can be rewritten as $$\frac{P[A \land B]}{P[B]} = ...
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Analysis of Digit Distribution in Powers of a Uniformly Distributed Random Variable
Let's get good enough (lower and upper) bounds for the sum, in closed form.
Put $f_k(x)=(x+1)^{1/k}-x^{1/k}$, so that (for $b$-ary system, in our case $\color{blue}{b=10}$)
$$
P(N_{m,k}=i)=b^{-m/k}\...
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Uncountable random graphs
Indeed as Brian points out measure theory does not play super well with uncountability. His answer shows that the theory of random graphs is not "categorical" in any uncountable cardinal.
...
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Secret Santa Perfect Loop problem
We're in the Hollidays Season and this play is quite usual here so I decided to spend some time on the problem.
Consider just a simplification on your rules. Remove rules number 3 and 4 and for any ...
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Why is it that probability density times interval width is equal to probability
I think I get it since $f(x)$ is never going to be less than zero, therefore when you take $f(x)$ times the very small $dx$ you are going to get $0 \leq f(x)dx \leq 1$. Remember now that probability $...
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Limit $\mathbb E[|(\text{Poi}(\mu)-\mu)/\sqrt{\mu}|^{k}]\, /\,\mathbb E[|\text{N}(0,1)|^{k}]$ as $\mu=k^{3}$ and $k\to\infty$
Let $X \sim \text{Poisson}(\mu)$ and $Y \sim \text{N}(0,1)$. Define:
$$
Z := \frac{X-\mu}{\sqrt{\mu}}, \quad
Z_k := \mathbb{E}[|Z|^k], \quad Y_k := \mathbb{E}[|Y|^k].
$$
We will show that, when $\mu$ ...
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Distribution function technique and exponential density
If $X_j$ is such that $$E(e^{-sX_j})=\int_0^{\infty}e^{-sx-\frac{x}{\theta_j}}\frac{dx}{\theta_j}=\frac{1}{1+s\theta_j}$$ and if $X_1,\ldots,X_n$ are independent, then assuming that the $\theta_k$are ...
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