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
32 views

Proof of $P(A'\mid B)=1-P(A\mid B)$ [duplicate]

Could someone help me understand how to prove $P(A'\mid B) = 1-P(A \mid B)$? I tried to make it so: $P(A'\mid B)= \cfrac{P(A'\cap B)}{P(B)}$ but I'm not sure how to continue. (I see that there is a ...
0
votes
1answer
17 views

Repdigit sequences

Is there a formula to determine the probability of a sequence of repdigits in a longer sequence of random numbers? The Feynman point in $\pi$, for example, occurring within the first $1{,}000$ ...
0
votes
1answer
16 views

Does the expectation for a blackjack hand become even in certain situations?

Particularly, when the player hand and the the dealer up card show the same value, shouldn't the odds of winning be about even at 50% (assuming infinite decks)? I ask this also because in some tables ...
0
votes
1answer
37 views

Probability of Passing Third Test After Failing the First Two

Homework Question: The probability that a person passes a test on the first try is $0.65$. The probability that a person who fails the first test on the second try is $0.75$. The probabilty that ...
0
votes
2answers
61 views

Probability in a fixed die

I have that transition matrix is ...
1
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3answers
109 views

$\mathrm E [X \mid X=x] = x$?

I've gotten so caught up in measure-theoretic probability that I'm actually having trouble showing this simple result. Let $X$ be an integrable random variable. Then $$ \mathrm E[X \mid X=x] = ...
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votes
1answer
25 views

Does conditioning reduces conditional variance i.e. $Var(W|Y) \le Var(W|Y,Z)$ [closed]

Let $W,Y,Z$ be are be some random variables. My question is does conditioning reduce variance on in other words is the following inequality true? \begin{align*} Var(W|Y) \le Var(W|Y,Z) \end{align*} ...
-1
votes
1answer
26 views

Probability question of winning in certain dice games [closed]

I would like to borrow your wisdom to solve probability of winning in these games 1: Dice blackjack Player has to get as close to 21 as possible with two dices rolled at a time. Player can stop ...
2
votes
2answers
56 views

Distribution of a convolution.

Assume that $X_1,X_2,X_3,X_4$ are IID such that $P(X_1=0)=0.3, P(X_1=1)=0.1$ and $X_1$ has on $(0,1)$ the density $f(x)=0.6$. Calculate $P(X_1+X_2+X_3+X_4 \leq 1).$ My work so far. It seems that ...
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votes
0answers
39 views

Finding distribution from PGF not in closed from.

$X_1,X_2,\ldots,X_N$ are independent and identically distributed random variables. We have $X = e^{-Y}$, where $Y\sim\mathrm{Poisson}(\lambda_u)$, and $$Z =X_1+X_2+\cdots+X_N ,$$ where $N \sim ...
1
vote
3answers
44 views

Trying to understand Bienaymé formula

In Bienaymé formula, it states that $var(\bar X) = \large\frac{\sigma^2}{n}$. However, when I was going through the proof here, it says the variances of $X_1,X_2,X_3......X_n$ are the same(assuming ...
0
votes
4answers
33 views

A probability question about drawing balls

"A box contains 2 red balls and 4 yellow balls. If 2 balls are randomly chosen and simultaneously removed from the box, what is the probability that only yellow balls are left in the box?" My work: ...
0
votes
1answer
23 views

Recovering density parameters from distribution function

Let $X$ be a random variable with probability density function $g(x;\theta_1,\theta_2)$, where $g$ is parameterized by two real numbers $\theta_1$ and $\theta_2$. I'd like to specify that $$ P(a \leq ...
2
votes
2answers
66 views

How to solve “ways of seating around a circular table”

Recently I asked a question about seating, here it is again: The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five ...
1
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0answers
21 views

How to bound $E \left[\left(E[Z^2\mid Y] \right)^2\right]- 2E \left[ |E[Z\mid Y]| \sqrt{E[Z^2\mid Y]} \right]$

I am looking for an upper bound on the following quantity \begin{align*} A=E \left[\left(E[Z^2\mid Y] \right)^2\right]- 2E \left[ |E[Z\mid Y]| \sqrt{E[Z^2\mid Y]} \right] \end{align*} where $Z$ is ...
0
votes
0answers
18 views

Chernoff-type bounds for Markov chains

I found the following result adapted from "Chernoff-type bound for finite Markov chains" by Pascal Lezaud, The Annals of Applied Probability, 1998, Vol. 8, No. 3, 849-867. Theorem: Let $P$ be the ...
4
votes
3answers
69 views

The probability of two consecutive non-leap years having 52 Fridays each is $\frac{5}{7}$. How?

I took a test on probabilities, and there was this question about finding the probability of two consecutive non-leap years having 52 Fridays each. I figured it would be $\frac{6}{7} \times ...
0
votes
0answers
25 views

How to get joint probability from Bernoulli correlation matrix (marginal parameters known)

Let's say I have $N$ Bernoulli random variables, $\left\{ X_{i}\right\} _{j=1}^{N}$, $X_j\sim Bernoulli(p_j)$ , and a correlation matrix $\mathbf{P}$: $$ \mathbf{P}=\left[\begin{array}{cccc} 1 & ...
0
votes
0answers
20 views

Formula of cumulative probability. [closed]

going to the question: let's say we have a game with 100 levels and ending one level produces a % to get a key. Once the key is obtained it cannot be obtained again so it is useless to continue ...
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votes
1answer
12 views

Minimizing expected loss based on asymmetric losses

I came across this problem in a book and am not sure how to approachi it: A dart will hit the random point $Y$ in $(0,1)$ according to the density $f_{Y} (t) = 2t$. You must guess the value of $Y$ ...
0
votes
2answers
54 views

Random increment through a probability distribution function

To Clarify i am trying to generate a random variable from a gamma pdf If $\Delta X$ indicates a random increment and it is said that $\Delta X$ follows a Gamma distribution. What would that mean ...
0
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0answers
34 views

Discrete Mathematics help [closed]

Suppose that a committee of 4 is to be chosen from 5 married couples. In how many ways can this be done? Of the possible committees how many contain exactly 2 women? Of the possible committees how ...
2
votes
1answer
37 views

Independence, conditioning, and correlation part 2 [closed]

Suppose $X$ and $Y$ are independent random variables. I now want to consider conditioning on some event $C$. Under what conditions will $X\mid C$ and $Y\mid C$ be correlated?
0
votes
1answer
34 views

Sum of two truncated normaly distributed variables

Let $X$ and $Y$ be two variables which are truncated normally distributed above zero (that is $X$ and $Y$ have the lower truncation point zero, their values are bounded above zero). Is $X+Y$ truncated ...
0
votes
0answers
57 views

A weird problem on expected value of a random variable [closed]

Let $X$ be a discrete random variable taking values $x_1, x_2$, ... with probabilities $p_1, p_2$, ... respectively. Then the expected value of this random variable is $E(X)=\sum_{i=1}^{\infty }x_i ...
1
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2answers
73 views

How many ways to arrange the seating?

The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with ...
0
votes
1answer
22 views

Probability with intersecting normal distributions

There are two independent random variables $a$ and $b$, each distributed normally with their own parameters. Given the means and standard deviations for $a$ and $b$, how can I calculate $P(a < b)$? ...
1
vote
2answers
32 views

Reduce Combination Formula

Hey i have to write a code for this: You can refer here: Picking Same Color Probability For the entire question. $\Pr(Success)=$$\sum\limits_{k=1}^{\min(m,n)}\frac{{m\choose k}\cdot{nm-m\choose ...
0
votes
1answer
35 views

Probability of decision on arrival of “new” information

I just decided to share a new problem (Rice's book, problem 1.58) and supposed solution: A teacher tells three boys, Drew, Chris and Jason, that two of them will have to stay after school to help ...
1
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1answer
18 views

expected squared prediction error derivation

I'm having a hard time deriving the formula on page 223 of Hastie et al. for the expected squared prediction error: Here are my first steps: $$ Err = E[(Y-\hat f(x))^2] = \\ E[(Y -f(x) +f(x) ...
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votes
1answer
56 views

How to find the inverse of $f(x)=2x - x^2$? [closed]

What is the inverse of $f(x)=2x - x^2$ in the domain (0,1) and the range (0,1)?
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2answers
33 views

How to find $P(X+Y \geq c)$ cumulative probability distribution

Suppose X and Y are both exponentially distributed with the same $\lambda$, what is the cumulative density distribution for the sum of the two? i.e. how do we find $P(X+Y \geq c)$
0
votes
1answer
51 views

For zero-mean r.v. $X$ with var. $\sigma^2$, want to show $E[e^{2X}]\leq e^{2\sigma^2}$.

Let $X$ have zero mean, $E[X]=0$ and finite variance $E[X^2]=\sigma^2<\infty$. I'm trying to show $$ E[e^{2X}] \leq e^{2\sigma^2}. $$ I started out with this related question, but I hadn't quite ...
1
vote
1answer
36 views

The equation to find chance that something will happen in a game

I'm trying to calculate the chance that an item will drop in the game I'm playing, or a similar event will happen. For a while I thought 100 - (chance it won't happen)^n * 100, was the corrent ...
2
votes
2answers
76 views

Probability returning to initial state

Let $P=\begin{bmatrix}0&\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&0&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}&0\end{bmatrix}$ and $P^{(n+1)}=P^{(n)}P.$ I know that if you start in any ...
1
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0answers
11 views

Finding probability density of $Y=X^2$, assuming $X\sim U(-1,1)$ [duplicate]

Suppose $X$ has a uniform distribution on $(-1,1)$, and let $Y = X^2$. How do we find $f_Y$?
1
vote
2answers
58 views

2 People Born on February what is the chance that they share the same birthday

I saw some similar questions to this floating around, but I don't think they are quite the same as this one. If you ask two people what month they were born in, and they respond February, what is the ...
2
votes
0answers
22 views

Expected response time of Continuous time Markov chain

I'm studying CTMC (Continuous Time Markov Chains). I came across the following slide I don't understand how they got $M(t+h) = M(t) + \alpha h + M(t)\lambda h - M(t) \mu h +o(h)$ Could anyone ...
5
votes
1answer
38 views

martingale and expectation

The following is an old exam problem: Let $\{X_n\}$, $n\geq0$, be a process adapted to a filtration $F_n$. Prove that $(X_n,F_n)$ is a martingale, if and only if for all bounded $F_n$-stopping time ...
1
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0answers
42 views

On the size of rational numbers and Irrational numbers. [duplicate]

Being a high school student, It's obvious to me that there are both an infinite number of rational and irrational numbers. However I don't really see if there is more rational than irrational, ...
1
vote
1answer
22 views

Conditional probability and disjoint events

If $\cup_{n=1}^\infty B_n=\Omega$ and $P(\Omega)=1$ then $\sum_{n=1}^\infty P(B_n)=1$, now $$P(A)=\sum_{n=1}^\infty P(A|B_n)P(B_n)=p\sum_{i=1}^\infty P(B_n)=p$$ If $X$ and $Y$ are independents ...
1
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2answers
53 views

Ten marbles put in a box, colour of each set by toss of a fair coin. You draw (with replacement) ten white marbles. Probability all marbles are white?

The following question comes from the probability section of the Titan Test*. * I will avoid the debate around whether this test accurately measures what it aims to, nor whether such aims are ...
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0answers
28 views

Probability help. Shared birthdays. [closed]

Suppose 10 people are in a room. What is the probability that there is at least one shared birthday among these 10 people? Express your answer in decimal form.
1
vote
1answer
18 views

How to derive formula for marginal probability of choosing nest in nested logit model?

I am trying to understand all the details of the nested logit and what confuses me is the formula for marginal probability of choosing the nest. In more details: the joint probability of individual n ...
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votes
4answers
32 views

Calculating probability of event $X$ happening $2$ out of $3$ times at $Y$ probability. [closed]

How do we calculate probability of event $X$ happening $2$ out of $3$ times at $Y$ probability? For example. What is the probability of team $A$ scoring two goals against their opponent if they were ...
1
vote
1answer
31 views

How can I convert [number of expected success per try] into [probability of succeeding N times without failing]?

There's a push-your-luck dice game called Can't Stop where you roll four six-sided dice, group the dice into pairs of your choice, then advance tokens along paths corresponding to the sums of your ...
1
vote
2answers
41 views

Geometric Probability with Lines

A point is on a number line, but it is limited to being at 0, 1, or 2. At each step, the point moves 1 unit left or right. If the point is at 1, it moves to either 0 or 2 with equal probability. ...
3
votes
3answers
47 views

Find $p_{ij}^{(n)}$ for the transition matrix

Let $$P=\begin{bmatrix}\frac{1}{3}&0&\frac{2}{3}\\\frac{1}{3}&\frac{2}{3}&0\\\frac{1}{3}&\frac{1}{3}&\frac{1}{3}\end{bmatrix}$$ find ...
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votes
1answer
21 views

Two coins and die are tossed simultaneously. [closed]

Two coins and die are tossed simultaneously. What is the probability that the number rolled on the die is less than three and a head and tail will show up on the coins?
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votes
3answers
42 views

probability question of a set of features

suppose we have set of events as follow: $$\{\emptyset, a, b, c, d, ab, ac, ad, bc, bd, cd, abc, abd, acd, bcd, abcd\}$$ whereas a b c and d are four different features that might be observed solely ...