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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3 Events, Conditional Independence

Given $A,B,C$ such that: $$ P(A|B),P(A|B^c),P(B|C),P(B|C^c) \mbox{ are known } $$ and that $A,C$ are conditionally independent given $B$, so that: $$ P(A|B\cap C)=P(A|B),P(A|B^c\cap C)=P(A|B^c) $$ ...
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28 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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3answers
34 views

Intuitive meaning of the probability density function at a point

I understand how to integrate probability density functions to find probability within a certain range. However, what I don't understand is what it would mean to set the variable (say x or y) to a ...
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1answer
29 views

What time should Celia aim for in her sixth race to make the team?

To be on the 1-km race team, Celia must have a mean time less than 5 min 50 sec in her 6 tryout races. Her times in 5 races are: 6 min 2 sec, 5 min 53 sec, 5 min 45 sec, 6 min, and 5 min 34 sec. What ...
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1answer
14 views

When randomly distributing n points amongst m people, what are the odds that one certain person will get a certain amount of points?

I'm mostly curious about how to find this in general, but the actual problem is with 20 points and 5 people. I know probability problems are very counterintuitive, and thus I was unsure after ...
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1answer
24 views

Probability with $n$ successes before $m$ failures

Independent trials resulting in a success with probability $p$ and a failure with probability $1 − p$ are performed. What is the probability that $n$ successes occur before $m$ failures? Given ...
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2answers
18 views

Mathematical Backing for Observations seen in Adding Independent Random Variables Together

So I have a function Y = F(N) that takes as an argument an integer number N and returns a summation of N sine-waves of different random parameters. I have plotted the results of two function calls ...
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0answers
15 views

Probability, expected frequency and resultant distribution skewed or not?

A population consisting of a certain proportion of defective items has mean $\mu = 2$. If a sample of 4 items is examined and repeated 200 times, obtain a) probability of an item being defective, ...
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2answers
27 views

defective component and probability

An electronics industry uses three plants from A, B and C in the ratio of $3$ to $2$ to $1$. However $1\%$ of components manufactured by A, $2\%$ of components manufactured by B and $3\%$ of ...
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1answer
23 views

Games and statistics

Three individuals A, B and C alternate in contention of a game according to the following rules: A plays with B and the winner plays with C. The game continues until one of the individuals to win two ...
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10 views

Coupon collector problem with $k$ distinct coupon sets to complete

In the standard coupon collector problem we have an urn with $n$ different coupons, from which coupons are being collected, equally likely, with replacement. Simple analysis shows that the expected ...
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1answer
20 views

Finding conditional probability using Baytes formula

Suppose that chips for an integrated circuit are tested and that the probability that they are detected if they are defective is $0.95$, and the probability that they are declared sound if in fact ...
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11 views

Maximum difference between tails in absolute value

I toss a fair coin $n$ times. Some notation: $S_i=$ difference between #heads and #number of tails after the first $i$ tosses, $1\leq i\leq n$. $M_n=max(S_1,S_2,\dots,S_n)$, ...
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1answer
46 views

Probability that AJ is guilty

A crime has been committed by a solitary individual, who left some DNA at the scene of the crime. Forensic scientists who studied the recovered DNA noted that only five strands could be identified and ...
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0answers
5 views

Concentration bounds on Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...
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1answer
43 views

Error in Billingsley?

Problem 8.25 in the third edition of Probability and Measure by Billingsley (1995, p. 142) is as follows: Suppose that an irreducible [Markov] chain of period $t>1$ has a stationary ...
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2answers
34 views

Probability that among 3 random digits two different one

I have been trying to solve the following problem: What is the probability that among 3 random digits, there appear exactly 2 different ones? The formula for no repititions is: ...
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1answer
43 views

Age distribution when meeting

I have a question regarding Poisson process. I will tell the story in the context of a player-monster game. Consider a player who is born at $t=0$. He will win the game if he can survive until ...
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2answers
34 views

Selecting k distinct numbers from an array with increasing probability distribution

I have to select k distinct numbers from an array such that probability of a number getting selected is more if it is at the end of the array (probability increases linearly). I'm thinking of ...
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1answer
29 views

Chessboard pawns arrangement clarification

I have a 8 X 8 chessboard, and 8 identical pawns. These pawns are arranged at random. What is the probability that the pawns are arranged in such a way that each row and column have only one pawn? My ...
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0answers
35 views

rolling a single die ten times

I have the following problem on a homework assignment for my Probability theory course: You roll a single six sided die ten times. What is the probability that you roll four 1's, three 2's, and three ...
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0answers
26 views

Conditional distribution

One point is chosen at random in the square $Q=\{|x| + |y| \leq 1\}$. Let $(X, Y)$ coordinates that point. a) The random variable $X$ and $Y$ are independent ? b) Find the density of $X$ given that ...
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2answers
23 views

Simplifying with Summation

This is a problem out of my statistics book but my issue is simplifying from Step 3 to Step 4 below: Step 1: var X=$\sum\:p_i\:(x_i-E[X])^2$ Step 2: var X=$\sum\:p_i[x_i^2+E[X]^2-2x_iE[X]]$ Step ...
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2answers
28 views

Understanding the geometric distribution

Simple question that has to do with the interpretation of the geometric distribution and frequency function: $P (X=k) = (1-p)^{k-1}p $ for $k = 1,2,3... $ where we are interpreting X as being up to ...
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1answer
41 views

Rumor and probability

31 people in a community, a person has a rumor to a second which, in turn, repeats to the third, etc. At each step the person receiving the rumor is randomly chosen among 30 people available. a) Find ...
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2answers
39 views

Two urns, A and B, each with two drawers.

If you have two urns, A and B, each with two drawers. The urn A has a gold coin in a drawer and a silver coin in the other drawer while the urn B has a gold coin in each drawer. An urn is chosen at ...
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1answer
55 views

How many steps would it take to get to the top of this staircase?

There are 26 steps in a staircase. You have a 51% chance to step onto the next step, and a 49% chance to step back down to the step prior. Assuming you are already on the first step, how many steps ...
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3answers
27 views

Dice roll - Geometric Distribution Question

I am having a hard time understanding the concept of a negative binomial distribution. For example the question: How many times do you expect to roll a six-sided die before landing on the number ...
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0answers
25 views

Uniformly boundedness of convolutions

Assume $X$ is an absolutely continuous random variable with pdf $f:\mathbb{R}\to[0,\infty)$. Assume further there exists $M>0$ s.t. $|f(t)|\leq M \quad\forall t\in\mathbb{R}$. Let $X_1,\dots,X_n$ ...
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0answers
18 views

How to transform 3 probability values to a specific range?

There are 3 probabilities say x, y and z such that x+y+z = 1. Now, we need to convert these three probabilities together in the range of 0 to 1. If x is 1 then it should be 0, if y is 1 then it should ...
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2answers
33 views

Deal 4 cards from a deck of 52 cards. What is the probability that we get at least one card that is diamonds?

Deal 4 cards from a deck of 52 cards. What is the probability that we get at least one card that is diamonds?
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1answer
19 views

Almost sure convergence through subsequences

$\{X_i\}$'s are independent Poisson random variables with parameters $\lambda_i$ respectively satisfying $\sum_{n=1}^{\infty}\lambda_n=\infty$. Define $S_n=X_1+X_2+\cdots +X_n$ then show that ...
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1answer
24 views

uniform distribution vs normal distribution for discount use case [on hold]

Problem statement: Reward a customer with lucky draw coupon of X% discount in between 1% to 100% Assume that slabs are pre-defined ( all are theoretical) 1% discount : 90% customers 10% ...
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3answers
39 views

Adding two random variables with convolution

I am trying to understand the purpose of convolution of two probability functions. Also when it is appropriate to use the convolve function on two independent probability distributions. ...
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1answer
18 views

Probabilty of Drawing Specific Cards

If I have a deck of randomized cards and I draw 5 cards, what is the probability that I will draw at least one 2 and at least one 3. In other words, I am looking for any hands of the form x2xx3, ...
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2answers
40 views

What is the distribution of a binomial variable where the number of trials is itself random?

We do the following experiment: Select a random element $k$ from $\{1,\dots,n\}$. Toss $k$ fair coins. Define $X$ = the number of heads. What is the distribution of $X$? Given $k$, the variable ...
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2answers
45 views

Probability: Number of guesses to get the “correct” item from a set, after repetitions

Suppose a friend has N overturned cups on the table, one of which (chosen by the friend at random) has a ball underneath, and you have to keep guessing until you get it right; whenever you guess the ...
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0answers
28 views

What is the intuition behind slovin's formula?

Here is the formula. $n=\dfrac{N}{1+Ne^2}$ I don't understand why this equation works. What value does $1+Ne^2$ represent?
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35 views

alarm clock question [on hold]

A student has a somewhat unreliable alarm clock. The alarm clock will ring at the pre-set time with probability 0.7. If it rings, the student wakes with probability 0.8. If it does not ring, the ...
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2answers
42 views

What is wrong with this logic based on a geometric distribution?

Problem #10 from A Collection of Dice Problems by Matthew M. Conroy, is: Suppose we can roll a $6$-sided die up to $n$ times. At any point we can stop and that roll becomes our score. The goal is ...
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1answer
35 views

“Well known properties” of Poisson distribution

I'm working with Bradley Efron (2010): Large Scale Inference and my question concerns the proof of Lemma 2.3. Here we have $z_i \sim F_0$ with probability $\pi_0$, $z_i \sim F_1$ with probability ...
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1answer
18 views

Conditionally independent and intersection

I'm trying to show that, given events $A,B,C,D$, such that $A,B$ are conditionally independent given $C$, whether or not $A,B$ are conditionally independent given $C\cap D$. I spent a couple of hours ...
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0answers
25 views

Combining a set of conditional probabilities

I'm interested in combining a set of conditional probabilities into one. For example, if I know the following probabilities: P(illness|male) ...
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2answers
28 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 ...
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0answers
12 views

Monotonicity of this conditional probabilty

$U_i ( i=1,...,n)$ are i.i.d and (0,1) uniformly distributed variables. Define its truncated product at threshold $\tau$ $(\tau\in(0,1))$ as $U=\Pi_{i=1}^nU_i^{I(U_i< \tau)}$. Define ...
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1answer
33 views

Finding Probability of picking one ball out of N balls.

presented with n identical balls, one with a prize in it. Picks each ball out idependently one at a time till gets prize. I need to find the mean and variance of the number of balls needed to pick ...
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1answer
23 views

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 ...
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1answer
48 views

{If $\frac{1+4p}{p},\frac{1-p}{4},\frac{1-2p}{2} $ are probabilities of$ 3$ mutually exclusive events.

If $\frac{1+4p}{p},\frac{1-p}{4},\frac{1-2p}{2} $ are probabilies of $ 3$ mutually exclusive events. Then, $$1) p=1/2$$ $$2) p=3/4$$ $$3) p=1/3$$ $$4) none of these$$ My Approach: ...
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2answers
34 views

Deck of $132$ different cards, $200$ draws with replacement

So we have a deck of $132$ (unique) cards, and we draw $100$ cards (with replacement). What is the $\%$ of unique cards that we can expect in the $100$ cards drawn? I started that question with the ...
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1answer
35 views

The probability of the student passing in test A,B,C are p,q and 1/2.If the probability that the student successful is 1/2.Then?

A student appears for tests A,B and C. The student is successful if he passes either in tests A and test B or tests A and test C. The probability of the student passing in test A,B,C are ...