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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Number of outcomes times probability of failure is number of failures?

I was hoping someone could spread some insight on why this is true? Suppose that in some situation, there are n! number of possible outcomes. Also, suppose that p is the probability of failure of ...
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1answer
260 views

Different Perspectives of Multinomial Theorem & Partitions

There are 2 important interpretations of the multinomial theorem and coefficients. 1: Determining the number of ordered strings that can be formed using a set of letters. For example, with 1 m, 4 ...
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1answer
11 views

uniform convergence of random variables

Consider a continuous function $f:[a, b] \rightarrow \mathbb{R}$. Let $U_{1}, U_{2}, \cdots$ independent and identically distributed with $U_{k}\sim U([a,b])$. Show that: $$ \dfrac{f(U_{1}) +f(U_{2}) ...
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2answers
17 views

Need help with some equivalent statements of measurability

I want to know why the above statements are true. Thank you!
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0answers
19 views

Probability of a sentence when billion monkeys are typing for 10 billion years

Suppose a billion monkeys type on word processors at a rate of 10 symbols per second. Assume that the word processors produce 27 symbols, namely, 26 letters of the English alphabet and a space. These ...
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2answers
19 views

Total probability

Tickets for games which USA plays on Basketball WC almost ran out. There are still 6 tickets for game with Spain, 8 tickets for game with Brazil and 12 for game with Ukraine. John buys two tickets. ...
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1answer
11 views

Every measure of natural numbers and the power of natural numbers as their sigma algebra looks like this…

Let X= $ \mathbb{N} $ ans S= P($ \mathbb{N} $) . Prove that every measure $\mu $ in $(X,\mathcal S)$ can be obtained by an unique non-negative extended sequence of real numbers $(a_{n})$ as follows ...
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6 views

simple probability questions

i) Of a group of patients having injuries, 28% visit both a physical therapist and a chiropractor and 8% visit neither. Say that the probability of visiting a physical therapist exceeds the ...
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0answers
12 views

How to prove the above Bonferroni inequalities using indicate function?

My problem is how to prove the above Bonferroni inequalities using indicate function?
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3answers
6k views

In need of tips/suggestions when to add or multiply probabilities

I am having trouble deciding when to add or when to multiply probabilities as in the following example. I know that by constructing Probability tree diagrams we could multiply along branches and add ...
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1answer
26 views

Conditional probability plane problem

I was presented with this problem and am not sure where to take it. A plane is missing and is presume to have equal probability of going down in any of 3 regions. If a plane is actually down in ...
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1answer
16 views

Compute Distribution of Linear Function

Let $Y = aX + b + \varepsilon $ , $X \sim \exp (\theta )$ , $\varepsilon \sim N(0,{\sigma ^2})$ , $X$ and $\varepsilon $ are independent How we can find distribution of $Y$? Thank you.
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0answers
55 views

Probability of at least m in a row out of n? (generic formula)

In a previously asked question of mine, I was specific in asking for a 75% freethrow shooter, what is the probability he would make at least 5 freethrow shots in a row out of 10. That means he would ...
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1answer
12 views

How to compute $P(X\leq Y)$ and $E(X^2 Y)$ with given probabilities [on hold]

Given $P(X=1, Y=0) = 0.1; P(X=1, Y=1) = 0.1; P(X=1, Y=2) = 0$ and $P(X=2, Y=0) = 0.2; P(X=2, Y=1) = 0.4; P(X=2, Y=2) = 0.2$ How do I compute $P(X \leq Y)$ and $E(X^2 \cdot Y)$?
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0answers
16 views

Ratio between normal distributed and gamma distributed variables

Let $X \sim N(0,1)$ and $G \sim Gamma(a)$. Why is $\frac{X}{G}$ t-distributed?
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1answer
24 views

How to prove this expectation equality?

How to prove this expectation equality? I am studying probability theory by myself and I find it hard. Thanks!
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0answers
4 views

average dirichlet distribution

Is it possible to combining 2 Dirichlet distribution averaging their values? The resulting probability distribution is still a Dirichlet distribution? If not how can I merge 2 similar Dirichlet ...
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2answers
29 views

Elementary problem in Probability

A disease is spreading across the city of Rome. If the vaccine is taken, there is $ 95/100 $ probability that you won't be infected. Independently, there is $ 25/100 $ probability that you won't be ...
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0answers
29 views

how can i get this probability expression?

A deck of $n=10$ cards is numbered from $1$ to $10$. The cards are shuffled and laid down from left to right, face up. Order each of the five successive pairs of cards. Each of these five pairs ...
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0answers
13 views

A binomial-ish probability calculation

The probability $p_n$ that $n$ customers visit a supermarket in one day is $p_n=p^nq, n\ge 0$ where $p+q=1$. Also, on an average two out of three customers buy a certain type of item. The probability ...
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2answers
180 views

What is this conditional probability asking?

I really dont understand how this info helps me solve this problem: In a study it was discovered that 25% of the paintings of a certain gallery are not original. A collector in 15% of the cases makes ...
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0answers
17 views

“Taking expectation” to yield conditional probability

This argument is taken from Resnicks Adventures in stochastic processes and let $T _{\infty } < \infty $ denote that an infinite number of transitions in a continuous time markov chain has occurd ...
3
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1answer
42 views

$P(X^2+Y^2<1)$ of two independent n(0,1) random variables

Suppose that X and Y are independent n(0,1) random variables. a) Find $P(X^2+Y^2<1)$ Attempt: a) Let $U = X^2 + Y^2$, $V = Y$. Then $X = \sqrt{V^2 -U}$, $Y = V$. $J = \left| ...
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1answer
22 views

Sum of $X_1,\ldots,X_n \sim \mathrm{exp}(1)$ i.i.d. random variables

Let be $X_1,\ldots,X_n\sim\mathrm{exp}(1)$ i.i.d. and $T_n =X_1 +X_2+\cdots+X_{n}$ Show with induction that the density of $T_n$ is $ \frac{1}{(n-1)!}a^{n-1}e^{-a}\,da $
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1answer
23 views

Even product on 5 dice rolls

A fair die (d6) is thrown five times. What is the probability that the product of the five scores is even? I have tried the following approaches. The first seems unnecessarily complex and also wrong: ...
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4answers
43 views

Question from probability theory

I have this question: A person has three children with at least one boy. Find the probability of having at least two boys among the children. EDIT* --> My intuition about the problem is this-- the ...
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1answer
12 views

Combinatorics Forming a word from set of Alphaphets

A sequence of 15 random draws, one at a time with replacement, is made from the set { A,B,C ...,X,Y,Z } of the English Alphabet(26 Alphabets in total). What is the probability that the string ...
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1answer
25 views

Ordered sequences of integer with fixed sum

Let $I_S = \{0, 1, \ldots, S\}$, with $S \geq 1$. Consider all the ordered sequences of length $L \geq 2$ in $I_S^L$ such that the sum of all the terms is equal to $S$. Let $N(L,S)$ be the number of ...
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0answers
22 views

Relationship between a distribution function and the truncated distribution function

Let $F(x)$ be a distribution function and $G(x)$ be $F(x)$ truncated on some interval $(a,b)$. I want to show that: $$G(x)=\frac{F(x)-F(a)}{F(b)-F(a)}, a<x \leq b$$ I want to do this by using ...
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1answer
34 views

some question about expected value

Let $X$ be a non negative random variable. We Know that $E(X)=0$. Is that correct that $X=0$ for some $X$. And more general: Is there a point in the probability space for which E[X]≤X and a ...
1
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1answer
33 views

Markov property question

In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions. However, I have the following statement ...
2
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2answers
41 views

Multiple Weighted Coin Toss Probability

1) Is it possible to weight two coins in such a way that if the two coins are tossed, the three possible outcomes (both heads, both tails, or one head and one tail) all have probability $\frac{1}{3}$? ...
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0answers
40 views

Ito integral solution

I need some help on how to compute the stochastic integral \begin{align} \int_{0}^{t}\frac{1}{\alpha-u}dW(u) \end{align} where $\alpha>0$.
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2answers
40 views

How to find $P(X>x)$ when the density is known but the integral does not seem to converge

I am trying to evaluate $$P(X>x) = \int_x^{\infty } t^{\kappa } \exp{\left(-\rho t^{\alpha\kappa + 1}\right)} \, dt$$ where $\kappa$, $\rho$ and $\alpha$ are all constants. I have tried some ...
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2answers
588 views

When do we use a two-tail hypothesis testing instead of a one tail?

I always use a two-tail hypothesis testing unless I am told to use a one tail. Is that a good way of going about solving problems or is there a flaw to that way of doing things? Also, if they say use ...
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1answer
36 views

How to prove the inequality using Jensen's inequlaity?

How to prove the above inequality? I am learning probability by myself and it has been confusing me for days. Thanks!
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0answers
30 views

about finding the probability when an event happens in competition

I am reading an interesting question about an ancient card game in China. I forgot the name but the rule is something like this. There are 1-50 participants playing the game each time. Each players ...
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3answers
29 views

How to prove the above expectation inequality?

If $\mathbb{E}[|X|^k]<\infty$ then for $0<j<k$, $\mathbb{E}[|X|^j]<\infty$, and furthermore $\mathbb{E}[|X|^j]\leq(\mathbb{E}[|X|^k])^{j/k}.$ How to prove the above expectation ...
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0answers
4 views

Measure to compare quality of synthetic data generated?

What is a good measure to compare the quality of the synthetic data generated with respect to the original data? The synthetic data I have, is the scaled up version of the original data. I am confused ...
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5answers
42 views

Counting principles question - team photo

A rugby team consists of 8 forwards, 7 backs and 5 subs. They all line up at random in one row for a picture. What is the probability that: a. the forwards are all next to each other? b. no two ...
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1answer
18 views

Probability of picking from a sublattice

Short version The set of partitions of a four-element set forms a lattice. Suppose that I pick $n$ times from the set of tri- and bipartitions (i.e., the top element = quadripartition and the ...
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3answers
86 views

Probability of impossible event.

There is question in my book: Probability of impossible event is? After reading the question my instant answer was $0$ and that was the answer given. But then i thought other way, question is ...
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1answer
14 views

Proving the Probability of a Subset

I am trying to learn probability and came across the following question. The conclusion proposed seems obvious but I am unsure on how to approach it. Any pointers in the right direction would be ...
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1answer
23 views

Probability when spinning a disk

A player spins a disk with three regions. The first region has the number 3. The second region has the number 5. The 3rd region has the number 4. On every spin, each of the numbered sections has a ...
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2answers
35 views

Find expected value of F(N)

If we are given that a variable X is defined as X=rand() % N Here rand() returns an integer between 0 and $10^{100}$ (inclusive) uniformly at random. Now we ...
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0answers
26 views

Dependent Expectation in Random Numbers Illustrated by Prime Repetition in Pi

When approximating Pi, appending each numerical digit as you refine, what is the first repetition of a four-digit prime number? For instance the first repetition of any one-digit number in the ...
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0answers
24 views

function of a random variable problem 2 [on hold]

Let $U$ be a continuous random variable with uniform distribution over $[0,1]$. Define $X$ by $$X=\operatorname{Int}\left(\frac{\ln(1-U)}{\ln(1-p)}\right)+1$$ where $\operatorname{Int}(x)$ is the ...
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1answer
39 views
+50

Monte Carlo p-test and early stopping

Say you have a coin with some probability $p$ of falling on heads. You would like to determine if this probability is less than or equal to $0.05$ with some reasonable degree of confidence and stop ...
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2answers
23 views

Consider the word 'PARTING'. What is the probability that a 4 letter sequence from this word contain the letter 'P'?

So far, I only have $_7C_4$ as the total amount of ways to arrange this (35). The answer is 4/7, and I can't really see how you get there.
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1answer
14 views

Probability - pmf for alternating probabilities

Two people, Amanda and Bella, will have consecutive arm wrestling matches until one of them wins two matches in a row and is declared the winner. Amanda wins a given match with probability ...