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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28 views

What is the PMF of the Hamming weight of a multinomial random variable?

Assume that $X$ is a random variable following a multinomial distribution of parameters $n$ (number of trials) and $p=(p_1,\dots,p_k)$ (event probabilities). Hence, ...
0
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1answer
32 views

Given 5 colors to choose from, how many ways can we color the four unit squares of a $2\times 2$ board

Given 5 colors to choose from, how many ways can we color the four unit squares of a $2\times 2$ board, given that two colorings are considered the same if one is a rotation of the other?
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3answers
32 views

Given a 50 card deck with cards numbered from 1 through 10 in each of 5 suits, how many

Given a 50 card deck with cards numbered from 1 through 10 in each of 5 suits, how many 5 card hands are there that include exactly one pair of two cards that have the same numeric value?
-5
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1answer
44 views

Probability of two heads in a sequence of coin flips [closed]

A fair coin is tossed six times and the sequence of heads and tails is recorded. What is the probability that the sequence contains exactly two heads? Express your answer as a common fraction.
2
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1answer
27 views

Intuition about Blumenthal's 0-1 law

I'm studying Brownian motion from Durrett. I'm trying to understand what Blumenthal's 0-1 law really says about what Durrett calls the germ field, $\mathcal{F}_0^+$. Let $\mathcal{F}_t^+ = \cap_{s ...
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0answers
26 views

Pure death processes

If $P_n (t)=\Pr (N (t)=n)$ and $N (0)=a$, how can I show that in a pure death process $$P_{(a-1)}(t)=a (e^{\mu t }-1)e^{-a \mu t}.$$ I showed that $P_a(t)=e^{-a \mu t}$. In fact I want to show ...
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votes
2answers
31 views

Central Limit Theorem and understanding mean for a single object

The IQ of actuarial science majors is assumed to be normally distributed with mean 112 and standard deviation of 14. In a class of 19 students, find the probability that the mean IQ of all 19 students ...
-1
votes
3answers
50 views

Probabilistic problem with balls in a Box. [closed]

A box contains 5 white balls and 6 black balls. Five balls are drawn out of the box at random. What is the probability that they all are white? Please help me ASAP!!!! Thank you!
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0answers
17 views

Struggling to understand multi-class logistic regression

It is well defined that given a data set of $N$ $i.i.d$ observations $\mathbf{X} = \{\vec{\mathbf{x}}_1, \dots, \vec{\mathbf{x}}_n\}$, along with corresponding target values $\vec{\mathbf{t}} = {t_1, ...
-1
votes
1answer
16 views

probability of predicting positions in a league [closed]

There are 20 teams in the premier league.how do I work out the probability of predicting the exact position of each team in the league
0
votes
1answer
23 views

Type I error in Normal distributions

Let $X_1,\dots , X_n \stackrel{iid}{\sim} N(\mu, \sigma^2 = 4)$ Test $H_0: \mu = 10$ vs $H_1: \mu > 10$ take a random sample of $n=16$ and reject $H_0$ if $\bar{x}>14$ Find $\alpha$ the type I ...
-1
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0answers
45 views

probability distribution of Complex Gaussian column vector and conditional probability of complex Gaussian column vector

I have column vector $\vec r=[r_1\ r_2]^T$. $$\vec r =hA\vec s +\vec n$$ where $h$ is a complex number , $\vec n \sim \mathcal{C} \mathcal{N}(\begin{bmatrix} 0 \\ 0 ...
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votes
1answer
41 views

Type I and type II errors

Let $X \sim uniform(0,\theta)$ we are testing $H_0: \theta = 1$ vs $H_1: \theta >1$ If we know that we reject $H_0$ if $X>0.9$ (1) find $\alpha$, the type I error (2)Suppose that $\theta=1.1$. ...
-3
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0answers
20 views

doubt in programming mathematics [closed]

anyone pls help me...how this 0.5625 probability comes from in between 5 and 3
-1
votes
1answer
33 views

Given circles with radius R1,R2(has same centre) what is probability of point which lies between R1 and R2 is closer to R2 than R1 [closed]

There are two circles that have the same center and are of radii $ R_1 $ and $ R_2$.We should place a point in the area formed between $R_1$ and $R_2$. What is the probability that the point is closer ...
0
votes
1answer
28 views

Probability of time between two events in a poisson process

Suppose people arrive at a certain place according to a poisson process with rate 10 per day. 1) What is the expected time until the arrival of 100 person. 2) What is the probability that ...
2
votes
1answer
71 views

Probability of getting the same vector result

This is part of a mathematical puzzle I was given to me by a friend a while ago and I can't work out how to solve it. Does anyone have any ideas? For a given vector $v \in \{-1,1\}^n$ we consider the ...
1
vote
2answers
71 views

Expected number of coin tosses [closed]

Consider a perfect coin with two sides (A and B) each equiprobable $P(A)=P(B)=P(\bar{A})=\frac{1}{2}$. What's the average number of tosses to get (AA) and (AB)? (all tosses independent) Closed form ...
-4
votes
1answer
39 views

Probability of atleast 2 people having a common birthday in a room of 32 people [closed]

In a room full of 32 people, what is the probability that at least two of them have the same birthday. I considered the different cases like this, 1: No two people have birthday on the same day 2: ...
3
votes
0answers
39 views

Can Monotone Class Theorem be easier to check than $\pi$-$\lambda$ Theorem?

I've been working on problem 14.4 in Billingsley's "Probability and Measure", which says: "Let $C$ be the set of continuity points of $F$. Show that for every Borel set $A$, $P(F(X) \in A, X \in ...
2
votes
4answers
169 views

How to understand $E(XY)$ intuitively

I have no trouble understanding $\displaystyle E(X)=\int xf(x)\,dx $ and $\displaystyle E(Y)=\int y f(y)\,dy$ As each $x$ multiplies the corresponding $f(x)$ and we take the integral of it to ...
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0answers
18 views

Aerospace engineering Probability book (graduate)

I am a probability student interested in learning the applications of stochastic differential equations and processes for aerospace problems. I am new to engineering but I can pick up the basic ideas. ...
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0answers
25 views

Probability of multiple wins in a series of evenly matched teams

24 evenly matched players are divided into 6 teams. (Evenly matched by virtue of golf handicaps). Team assignment is random. One player frequently wins, irrespective of which team he is assigned. I ...
1
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2answers
33 views

Confusion regarding Burke's theorem

Arrivals occur at rate $\lambda$ according to a Poisson process the service time have an exponential distribution with parameter $1/\mu$ in an M/M/1 queue, where $\mu$ is the mean service rate where ...
2
votes
1answer
46 views

Intuition for probability density function as a Radon-Nikodym derivative

If someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for all ...
0
votes
1answer
28 views

Simple Probability Inequality with Stopping Times

Suppose $U_1,...,U_n$ are independent random variable with $\mathbb{E}[U_i]=0$. Define $Z_k:=\sum_{i=1}^k U_i$. Set $T:=\inf \lbrace k \in N \mid |Z_k|>2\alpha \rbrace$. Clearly $\lbrace T=k ...
1
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0answers
31 views

A finite field subset sum count

Given $d\in\Bbb N$, pick $N=2^{2d}$ distinct $a_j$ from $\big\{1,\dots,2^{d^2}-1\big\}$ and pick $i$ from $\big\{3,\dots,2^{d}\big\}$. On average how many of $i$-subsets in ...
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votes
2answers
18 views

Question about finding a distribution without taking into account previous events

We have 8 prisoners, each has a probability of escaping (independently) each day of $0.4$, what is the distribution of the amount of escaping prisoners on the third day? This is the answer: the ...
1
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2answers
19 views

probability of rolling a die 15

if you roll a die 15 times, what's the prob that there are four 6's? Answer is $\binom{15}{4} * (1/6)^4 * (5/6)^{11}$ I am assuming the $(1/6)^4$ comes from the probability you get four 6's, and ...
2
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1answer
33 views

Coin flipping game with stop-loss

You play 100 rounds of a coin flipping game where you win \$2 for a head and lose \$1 for a tail on each round. Clearly since the coin tosses are independent the expected winnings are \$50. Now, ...
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1answer
39 views

probability of equiprobable events

suppose we have set of equiprobable events as follow: {∅,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 or coupled with ...
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0answers
44 views

When the sum of Markov chains is a Markov chain: “dumb” algorithm

Suppose I have two (independent) discrete-time and space, preferably non-homogeneous Markov chains $\Gamma^{(i)}=\{\gamma_1^{(i)},\gamma_2^{(i)},...\}, \ i=1,2$ and I want to find a way to check when ...
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0answers
22 views

Proving that the probability of meeting someone k times around n locations, is $P_n^k={1\over k!}{\sum_{m=0}^{n-k} {(-1)^m\over m!}}$

To be more specific on the question: Two people go around n locations on a random path, say $A, B, C...$, with steps between locations being the same in terms of time. That is that when, for example, ...
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0answers
19 views

Probability distribution to failure [closed]

I am going to do a simulation for a manufactruing system, i must consider a scenario as: a $20\%$ probability of failures occurring in $M1$. Q: What is the probability distributions the time to ...
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2answers
25 views

Interpretation of double factorial solution to a problem of pairing two types of objects

I'd like a clarification about or some insight into one possible form of solution to the following problem: Suppose that each of n sticks is broken, into one long and one short part. The 2n ...
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2answers
35 views

Finding the expected number of trials in an experiment.

Given a uniform probability distribution over $[0, 1]$, a number is randomly selected from this distribution. We have to find the expected number of trials such that the sum of the picked numbers $ ...
1
vote
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
38 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
62 views

Probability in a fixed die

I have that transition matrix is ...
1
vote
3answers
135 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] = ...
-1
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
27 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 ...
-1
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0answers
40 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
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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
68 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 ...