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
13 views

Writing the expected value of a random variable in terms of its cumulative distribution function

My professor said that an alternative expression for the expected value of a random variable can be written as: $$ E[X] = \int_{0}^{\infty} (1-F_X(x)) \, dx - \int_{-\infty}^0 F_X(x) \, dx $$ No ...
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0answers
3 views

Sampling substrings of a beaded necklace to determine the necklace composition

I have a necklace composed of 100 beads, where each bead is one of 13 colors. If I am only able to look at one 4 bead sub-sequence at a time (connected, as they would be on the necklace) , how many ...
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0answers
25 views

Polynomial Interpolation When part of $y_i$'s are Shuffled

Hypothesis: Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and ...
1
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0answers
27 views

Is this the Birthday problem principle?

I have some questions about the Birthday paradox. Let's say there is some positive integer n that is somewhere between 0 and N (also a positive integer). I tell the program to start generating random ...
0
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3answers
26 views

The dice is rolled 10 times and the results are added with given conditions.

Q: A dice has one of the first 6 prime number on each its six sides ,with no two sides having the same number .the dice is rolled 10 times and the results added.the addition is most likely to be ...
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0answers
25 views

Show that If E ⊂ F, then P(E) ≤ P(F).

I am trying to solve this following problem If E ⊂ F, then P(E) ≤ P(F). But i am having no idea where to start from. Can anyone please help me with that? Thanks
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0answers
14 views

how to find joint probability distribution under conditioning [on hold]

given two pdf of random variable X,Y. Whether it is possible to find joint pdf of x and y under the condition x>y. Is it possible to consider x and y are independent.
4
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2answers
34 views

Suppose $\xi_1, \xi_2,\ldots$ are i.i.d. random variables with mean $\mu$, variance $\sigma^2$. Form the random sum $S_{N} = \xi_{1}+\cdots+\xi_{N}$.

(a) Derive the mean and variance of $S_{N}$ when $N$ has Poisson distribution with parameter $\lambda$. So far, for the mean, I have the following: $E[S_{N}] = E[E[S_{N}\mid N=n]]$ $$ = ...
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votes
5answers
848 views

What is the expected length of the largest run of heads if we make 1,000 flips?

Is there a way to calculate on average, the maximum amount of times we can expect a coin to land heads during 1,000 flips? So the answer (and formula if one exists) I am looking for would be ...
3
votes
1answer
34 views
+50

Poisson events distributed uniformly in a given time

It is given that $4$ Poisson events occur between $12:00$ to $13:00 $ (interval denoted by T). Intuitively, Why the probability of each event to occur at time $t ...
0
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1answer
30 views

Lower bound for $\Pi(n)$ - viability of probabilistic theory

Can somebody check the validity of my arguments below, and tell me why its wrong or right? Consider the sequence of non-negative integers. Let $a_0=0, a_1=1, ..., a_i=i,...$ Divisiblilty of $a_i$ ...
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1answer
20 views

Probability -dividing into groups

In how many ways can 12 people be separated into 3 groups of 4 if the 12 comprises 6 pairs of partners? We must keep partners in the same group, but we do not distinguish between the group $(a, b, c, ...
0
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0answers
17 views

Difference between the “Hazard Rate” and the “Killing Function” of a diffusion model?

I posted this question on Cross Validated - but I think it applies here too. Also, it increases the chances of the question being answered. Link here If this is not acceptable - administrators ...
8
votes
4answers
441 views

Probability that a natural number is a sum of two squares?

Some natural numbers can be expressed as a sum of two squares: $$2=1^2+1^2$$ $$25=3^2+4^2$$ $$50=7^2+1^2$$ If one chooses a random natural number, what would be the probability that that number is a ...
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votes
1answer
21 views

Probability - conditional

The probability that bulbs are detected faulty if they are defective is 0.95 and the probability that bulbs are declared fine if in fact they are fine is 0.97. If 0.05 of the bulbs are faulty, what is ...
3
votes
3answers
148 views

Coin pair betting paradox (NOT!)

If we throw two fair coins then there are 4 equally probable possibilities: HH, TT, HT, TH. Suppose we can't see the result, but we can check one of those two coins. (doesn't matter which one) ...
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0answers
13 views

How to prove that the set of all exchangeable events is a sigma-algebra?

Let $ {X_n}_n $ be sequence of identical R.Vs Mark by S the set of all sequences available from it. An exchangeable event is $E\subset S $ which is not sensitive for finite permutations. ...
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votes
1answer
28 views

Expected number of customers sitting on correct places

In a shop customers are given a seat number before entering the shop in the order 1,2,3,...,n but after entering the shop they sit in a random order not related to their seat number. what is the ...
1
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0answers
18 views

The probability that two matrix vector products are equal

Consider a random $n$ by $n$ circulant matrix $M$ whose first row entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first ...
0
votes
2answers
46 views

dice probability - same 2 dice in 6 dice rolls

I have this simple probability problem that I am not sure I solved correctly. I am not interested in formulas, but rather the thought process of how to solve it. Suppose we roll six 6-sided dice that ...
0
votes
1answer
18 views

In how many ways that the letters of ENTERTAINMENT are arranged in a row where two Es are together and one is apart [on hold]

In how many ways that the letters of ENTERTAINMENT are arranged in a row where two Es are together and one is apart??
1
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3answers
40 views

Random variable with 2 distribution functions

Just a question here, Given a random variable $X$ defined in a probability space, is it possible to have more than one distribution function $F$ ?
1
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1answer
32 views

Probability that a year contains 53 Mondays

The question: Find a probability that a year chosen at random has 53 Mondays. Now I know in a non-leap year, probability of getting 53 Mondays is $\frac{1}{7}$ and in a leap year, probability of ...
1
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1answer
26 views

Determining bounds for change sum of continuous r.v.'s

I'm trying to understand how to determine the bounds when computing the sum of continuous random variables. Here is a sample question: X and Y have the following joint pdf: $f_{X,Y}(x,y) = 4xy, 0 ...
0
votes
1answer
32 views

Drunk Passenger Probability question [duplicate]

I don't know how to solve this question. A line of 100 airline passengers are waiting to board a plane. They each hold a ticket to one of the 100 seats on the flight. For convenience, lets say that ...
0
votes
1answer
15 views

Derivative of Poisson that approximates Binomial

Instead of a standard urn ball problem, I have many urns and balls. Many. One might say, a continuum of balls $B$ and urns $U$. The likelihood of a single urn having $x$ matches is, under the ...
2
votes
1answer
23 views

Law of a random variable (characterization)

If $X$ is a real random variable defined on $(\Omega,\mathcal{F},\mathbf{P})$ then there exist several characterizations of the law of $X$ being $\mu$ : $X \sim \mu$ if and only if for every ...
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2answers
305 views

Probability of picking specific balls

Suppose I have $20$ red balls in one box and $20$ blue balls in another box. There $12$ red balls and $7$ blue balls have stars on them. I randomly take out one red ball and one blue ball at each ...
5
votes
2answers
171 views

Biology: How to find the probability of randomly generating multiple, sequentially identical sets

If I randomly generate a substring (example "ATGCAGC") with equal probability (1/X where X=4) for each digit with length (L) digits: What is the formula for finding the probability (P) of randomly ...
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2answers
16 views

how Find the probability that the committee will consist of the following all dentists

A committee of four people is to be formed from six doctors and eight dentists. Find the probability that the committee will consist of only dentists
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votes
1answer
54 views

Probability of randomly selecting one student from each of three cities [on hold]

The geographical distribution of hometown of some 80 students at DLSU-D is given as: 50 from Cavite, 10 from Laguna, and 20 from Manila. Suppose three students are selected. Find the probability that ...
0
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1answer
28 views

Question related to Square Integrable Martingale

Let $X_n$ and $Y_n$ be martingales with $EX_n^2<\infty$ and $EY_n^2<\infty$ for all $n$. Show that $EX_nY_n-EX_0Y_0=\sum_{m=1}^nE(X_m-X_{m-1})(Y_m-Y_{m-1})$. I tried to expand the right ...
2
votes
1answer
74 views

Monty Hall problem again (from Grimmet and Stirzaker)

Grimmet and Stirzaker Exercise 1.4.5.2 In a game show you have to choose one of three doors. One conceals a car, 2 conceal goats. You choose a door but the door is not opened immediately. Instead ...
1
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1answer
22 views

Expected value of function of minimum between two random variables

Two independent random variable $X,Y$ are distributed on $[0,\infty)$ according to the cumulative distribution function $F(x)=1-(x+1)^{-2}$. Let $Z=\min(X,Y)$. Determine $E\left[\frac{Z}{Z+2}\right].$ ...
0
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1answer
19 views

5-Card Poker Two-Pair Probability Calculation

Question: What is the probability that 5 cards dealt from a deck of 52 (without replacement) contain exactly two distinct pairs (meaning no full house)? Solution: ...
0
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2answers
26 views

Anagrams contained within random strings

What is the probability that a random string of length $n$ will contain an anagram of a shorter string of length $k$? E.g., you generate a string of 50 random letters, repetitions allowed, what are ...
1
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0answers
20 views

An inequality related to supermartingale?

Let $X_n\ge0,n\ge0$, be a supermartingale. Show that $CP(\sup X_n>C)\le EX_0$. I tried to use the inequality supermartingale satisfies, which is $E(X_n|\cal {F_{n-1}})$$\le X_{n-1}$. However, ...
0
votes
1answer
616 views

Expected state of a Markov chain

Let's start with a slightly trivial Markov chain defined as follows: the beginning state is called $1$ and the set of states is $\mathbb{N}$. At each step, when the current state is $n$, the ...
0
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0answers
28 views

How can I formed as below permutation problem

Hi I am writing a program and i encouraged the below permutation problem and need your help. There are 4 boxes: 3 of them have 2 balls The one box has 1 balls. The question is what is the ...
0
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0answers
27 views

Probability distributions associated with Markov chain

Let's say I have a Markov chain, with all the transition probabilities known, and there's a cost associated with each transition. The cost for transitioning from node $a$ to node $b$ is given by the ...
3
votes
2answers
32 views

characteristic function implies degenerate distribution

Let $X$ be a random variable with characteristic function $\phi(t)$ satisfying $|\phi(t)|=1$ for all $|t|\leq 1/T$ with some $T>0$. Show that $X$ is degenerate. i.e there is $c$ such that ...
0
votes
1answer
25 views

Prove that X has a chi square distribution

If $X_1,\dots ,X_{30}\sim N(1,\sigma^2)$ and $\hat \sigma^2 = \frac{\sum(X_i-1)^2}{30}, $ then show that $30\,\hat σ^2 /σ^2$ has a chi-square distribution with $30$ degrees of freedom.
2
votes
1answer
22 views

An Example of sequence of R.V with $E(X_n) = X_0$ but $E(X_n^{1/2}) \to 0$

I need an example of $\{X_n\}_n$ be a sequence of nonnegative, random variables, with the same finite expected value $E(X_n)=\mu_0$, that obeys: $E(\sqrt{X_n})>E(\sqrt{X_{n+1}})>\dots \to 0$
0
votes
1answer
24 views

Interpretation of the negative binomial and geometric distributions

I am having trouble putting together the way these distributions work. It doesn't matter whether we speak of the support space in terms of number of trials or failures. Specifically what variable is ...
5
votes
1answer
384 views

Probability that a given Poisson variable samples greater than its mean $\lambda$, provided $\lambda > D$

Given a random variable $X \sim \text{Poisson}(\lambda)$ such that $\lambda > D$, with $\lambda, D \in \mathbb{N}$, what is the probability that a sample obtained from $X$ is greater than ...
0
votes
1answer
27 views

How to calculate the odds of a 5x5 Bingo game?

I don't have a mathematics background, but am trying to calculate what the theoretical odds of winning a 5x5 bingo game is if 5 numbers are drawn. Eg board: ...
2
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0answers
17 views

a conceptual question on markov chain [duplicate]

Suppose $\{X_n,n\ge 0\}$ and $\{Y_n,n\ge0\}$ are two independent discrete-time markov chains (DTMC) with state space $S=\{0,1,2,\ldots\}$. Prove or give a counterexample to: $\{X_n+Y_n,n\ge 0\}$ is ...
2
votes
3answers
64 views

How do mathematician make sense of “outcome” and “events” in probability?

One of the biggest challenge for me to understand probability is to make sense of this concept of outcomes and events. To put it plainly, it just doesn't feel like mathematics anymore when we talk ...
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4answers
1k views

Minesweeper probability

I ran into the situation pictured in the minesweeper game below. Note that the picture is only a small section of the entire board. Note: The bottom right 1 is the bottom right corner tile of the ...
1
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1answer
355 views

Distribution of sum of multiplication of i.i.d. exponential random variables.

I have two questions: A) Suppose that we have $Z=c\Sigma (X_i-a)(Y_i-b) $ where $X_i$s and $Y_i $s are independent exponential random variables with means equal to $\mu_{X}$ and $\mu_{Y}$ (for ...