Questions tagged [elementary-probability]

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Probability of event A and B

Consider a events A (Man passes in exam A) P(A)=0.7 Event B (Man passes in exam B) P(B) =0.8 Let's take that exams are not related to each other and thus events are independent. Then what is the ...
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1answer
15 views

Basic proof for conditional expectations

$X$ is a random variable, $E[\mid X\mid ]<\infty$ and $F_X$ is its cumulative distribution function. I would like to proof that $E[X\mid X\geq b]\geq E[X \mid X \geq a]$ for $b\geq a$ and $F(b)<...
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3answers
22 views

Function of a random variable is a random variable

Consider the following two statements Statement 1 If $X$ is a continuous random variable and $Y=g(X)$ is a function of $X$, then $Y$ itself is a random variable. Statement 2 A random ...
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2answers
42 views

Car Convoy Problem [closed]

Cars travel down a one-way single-track road. The nth driver would like to drive at speed $V_n$ , where $ V_1,V_2, ..., V_n$ are iid random variables. Cars will get bunched into convoys. If $V_2>...
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1answer
42 views

Proofs that impossible event B has NO conditional probability $\mathbb{P}(A|B)$. Is everything correct?

"$A$" is an event that has probability somewhere in interval $(0,1)$. "$B$" is an event that has probability equal to zero. Given that, I will prove that $\mathbb{P}(A \mid B)$ doesn't exist. The ...
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1answer
34 views

Conditional probability question about children

Sushma has $2$ children. Let $A$ be the event that the oldest child is a girl, $B$ be the event that at least one child is a girl, and $C$ be the event that both children are girls. Which of the ...
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1answer
20 views

Coin flips; heads and two tails percentages [closed]

I have some simple rules and want to compute the probability either as a real number or as percentage of the outcomes. Say someone flips a coin, and there is $50$/$50$ chance or equal probability it ...
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1answer
82 views

Does $\{\} = \{\{\}\} \cup \{\{\}\} \cup \cdots \cup \{\{\}\}$

Does any finite set $S = \{a, b, ..., c\}$ equal the union of it's elements? I believe so, but would I write that like $$S = \{a\} \cup \{b\} \cup \cdots \cup \{c\} \tag{A}$$ $$\text{vs} $$ $$S = a \,...
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1answer
35 views

What is the best number of questions to guess for a multiple choice question consisting of 4 options?

There are 16 total questions and each wrong answer is worth -0.25 points, every correct answer is 1 point. An unanswered question is worth 0. Is it better to guess 5 questions rather than 4 assuming ...
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2answers
63 views

Expected value if two coins are drawn simultaneously from a bag containing $2$ one unit and $3$ two units of money

This is a question from the Resonance DLPD Algebra book (Page 250). To quote the book: From a bag containing $2$ one rupee and $3$ two rupees coins, a person is allowed to draw $2$ coins ...
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2answers
30 views

Conditionality Notation, Language, Concepts.

I've learned my entire 2 days of probability theory via wikipedia. So this is bothering me: how is it that $$P(A | B) = \frac{P(A\cap B)}{P(B)} \tag{1}$$ and two events are called independent when ...
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1answer
37 views

Simple example of a distribution on +ve natural numbers with $\sum_{n=1}^\infty n^2 p_n < \infty$ and $\sum_{n=1}^\infty \sqrt{p_n} = \infty$

DISCLAIMER: This is probably too elementary and I haven't spent much time thinking about it (as I should). Question What is a simple example of a distribution $(p_1,p_2,\ldots,p_n,\ldots)$ on +ve ...
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1answer
57 views

Probability that 4 single digit numbers whose sum is 28 contain at least two prime numbers

What is the probability that a list of 4, not necessarily distinct, single-digit numbers, whose sum is 28, will contain at least two prime numbers? There are not a whole lot of possible "desirable" ...
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2answers
136 views

Out of 11 tickets marked with nos. 1 to 11, 3 tickets are drawn at random. Find the probability that the numbers on them are in AP?

I kinda tried it and got the answer 5/33. But I feel like I am doing something wrong. What I did was:. Firstly I found favorable outcomes. My favourable outcomes were 25 such triplets. I think I am ...
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1answer
21 views

Why are these sets equal? From Morris H. DeGroot and Mark J. Schervish: Probability and Statistics, theorem 1.5.2.

"Proof. Consider the infinite sequence of events $A_1, A_2, \dots,$ in which $A_1, \dots, A_n$ are the $n$ given disjoint events and $A_i$ = $\emptyset$ for $i > n$. Then the events in this ...
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1answer
58 views

Probability remains same Why?

A box has $m$ black and $n$ white balls. A ball is drawn at random and put back with $k$ additional balls of the same colour as that of the drawn ball. Now a ball is drawn again. Find the probability ...
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2answers
28 views

Probability of objects are being sorted

There are 20 toffees (10 of mint, 10 of orange) in two boxes. I mix them together. Then I put 10 randomly chosen from the pot into one box and 10 in the other box. What is the probability that I end ...
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2answers
38 views

How many people need to turn up to be sure of making at least one team?

So pigeonhole principle states when there are m objects to be divided into n sets then at least one contains r+1 objects. i.e m > nr In this question the m objects should be 12 months in 5 people so ...
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2answers
61 views

Out of r digits chosen at random, what is the probability that no digits exceed k. (Range: $0 \le k \le9$.)

So far, I have concluded that my sample space obviously ranges from 0-9. I know that the digits can be chosen at random with replacement. Now the outcome probability of each digit being chosen is 1/9. ...
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1answer
14 views

Counting Size of Event Spaces, card drawing

Attached is a photo of the problem I'm having a hard time with. We are trying to calculate the likelihood of drawing an ace given 10 or more draws with no aces drawn. We are given two methods of ...
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1answer
66 views

What is the probability that the second card is a king, given that the first card was a spade?

I have the following questions. Part a- the answer is 4/51 as all the jacks are still in the pack. My question is why the answer to part b not the same as all the kings are still in the pack? Instead, ...
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1answer
36 views

Given the density function, show that $c sin^{-1}(\sqrt{x})$ is the distribution function and find c.

I was able to find c which is $F(x) = c sin^{-1}(\sqrt{x})$ if $0 < x < 1$ $F(1) - F(0) $ $c (sin^{-1}(\sqrt{1})-sin^{-1}(\sqrt{0})) = 1$ $c (\frac{\pi}{2} - 0) = 1$ $c (\frac{\pi}{2}) = ...
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3answers
29 views

How $P[X=x]= \frac2{3^x}$ can give an even value for $x =1,2,3,\dots$

I have a question that says “Let $X$ be a discrete random variable with probability function $P[X=x] = \frac2{3^x}$ for $x = 1,2,3,\dots$ What is the probability that $X$ is even? The thing is I don’...
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1answer
87 views

A box contains $n$ white and $n$ black pebbles. $r$ pebbles are taken out. Probability that the chosen pebble is black?

A box contains $n$ white and $n$ black pebbles. A fair sided die with $n$ sides(numbers from $1$ to $n$) is thrown and the number you get, let's call it $r$, is how many pebbles are taken out randomly....
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1answer
26 views

How to calculate the following probability?

There are $N$ coupons numbered $1,2,\dots,N$ in a box. If $n$ coupons are taken at random, what is the probability that $M$ will be the highest number on a ticket if the drawings are made with ...
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0answers
37 views

Probability of an element in continous sample space

Let us consider a continuous sample space $S$ of some random experiment; Is it true that $p(\{e\}) = 0$ for all $e \in S$ even in the case of non-uniform probability function?
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204 views

Number of ways of selecting $k$-element subsets from an $n$-element set?

I want to find the number of ways of selecting $k$-element subsets from an $n$-element set. Suppose we select the elements one by one. There are $n$ ways of selecting the first one, leaving $n-1$ ...
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3answers
94 views

What is the chance that a 1 in 1000 event will happen within the first 500 tries?

Say you have a thousand-sided die. You win if the die lands on 27, so a one in one thousand chance. I was curious as to what the chances are of winning if you roll the die 500 times. My gut told me ...
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1answer
34 views

Routine Hypergeometric Question, alternate way to solve?

I feel like my reasoning and solution for this question are satisfactory, but it was a multiple choice question with explicit decimals given for the options and the expression I arrived at for a ...
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2answers
46 views

Generating digits $0$ to $9$ with a coin

You're given a fair coin. The goal is to use it to generate a random decimal digit. I didn't find any solutions on the internet, so I came up with one myself. It goes as follows: Assign $0$ to ...
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3answers
150 views

How to label the events in a probability problem?

I'm starting a statistics and probability class and it's still pretty hard for me to "name" my probability events, which I think makes it very difficult to solve problems. I'll pick an example from ...
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1answer
90 views

What are the expected winnings of buying one ticket (including the cost of the ticket) if a ticket costs $\$3$ and the jackpot is $\$250000$?

In a lottery, a ticket costs $\$3$, and the jackpot is worth $\$250,000$. In all, $100,000$ tickets are sold, one ticket is randomly drawn, and the winner is awarded the jackpot. What are the expected ...
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0answers
36 views

Basic conditional probability with ball sustitution

i'm working on some basic conditional probability excercises (too much time not doing this math discipline), and I don't have the solution for this one. I think that is correct, but just want to ...
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0answers
31 views

A question related to Probability(read body)

Q. Out of 100 students two sections of 40 and 60 students are to be formed. If Jack and Jenny are among the 100 students, what is the probability that: a) they end up in the same section? b) they end ...
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1answer
56 views

Writing sample space and events as sets explicitly

Let us consider the following question Devasena can select either Baahubali or Bhallaladeva but not both. If she selects Baahubali, then she will be happy with a probability $\dfrac{9}{10}$;...
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2answers
72 views

Probability problem solved in two ways. Can someone please conclude which method to use in which case

I found an elementary problem on probability that left me a bit confused- 6 married couples are standing in a room. If 4 people are chosen at random, then the chance that exactly one married couple ...
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2answers
68 views

Probability question of rolling 6 dices and getting required events

Question: Samuel Pepys wrote Isaac Newton to ask which of these events is more likely: that a person get (a) at least $1$ six when $6$ dice are rolled. (b) at least $2$ sixes when $12$ dice are ...
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2answers
704 views

Basket with Lilies and Roses. Find the number of roses.

Ques: A covered basket of flowers has some lilies and roses. In search of rose, Sweety and Shweta alternately pick up a flower from the basket but put it back if it is not a rose. Sweety is 3 times ...
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1answer
50 views

Determining an unknown value c for a sum of iid random variables [closed]

$X_1, X_2, and X_3$ are independent random variables with $X_k \sim N(0,k)$. (So, for example, $X_2$ is a normal variable having mean 0 and variance 2). Determine the value of c such that $$c(12X_1^2+...
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1answer
131 views

Finding the expected value of a variable using double integration in polar coordinates.

This is a statistics problem, but it requires integration. And I'm getting a little lost in the problem because there are multiple variables involved when you switch over to polar coordinates, and I ...
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1answer
19 views

Probability Distributions $P(A^c\cap B^c)$? [closed]

How can i do this distribution from Pitman probability textbook? I dont know how this expands. Do i use the compliment rule? Difference rule? Inclusion-exclusion?
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1answer
24 views

Given a probability $p$, what is the upper bound of how many columns in a row-stochastic matrix exceed $p$?

Given a probability $p$ and a row $i$ of a stochastic matrix $M$, what is the maximum number of columns $j$ such that $M_{ij} \gt p$? ...
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1answer
73 views

Probability that two random numbers have a Sørensen-Dice coefficient over a given threshold

The following problem came up at work and my probability knowledge isn't up to the task. Let $a, b \in \mathbb{Z}_{2^n}$ be two $n$ bit integers. Their Sørensen-Dice coefficient is the quantity $$ DS(...
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1answer
53 views

probability problem solution with full proof [closed]

n+4 apples are distributed at random among n persons. Find the probability that atleast one of them will receive none. I want a detailed solve using complex combinatorial summation. Please help me to ...
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1answer
39 views

Conditional probability as a function

Sample space (S) is a set of all possible outcomes of a random experiment. $S = \{x | x $is an outcome$\}$ Event (E) is a subset of sample space. $E \subseteq S$ Probability is a function from ...
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1answer
30 views

Probability of Random Selection [duplicate]

Fifteen coupons are numbered 1 to 15. Seven coupons are selected at random, one at a time with replacement. What is the probability that the largest number appearing on a selected coupon be 9? My ...
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0answers
23 views

Probability that the drawn ball is the only ball of its kind.

A bag contains 7 balls of any colour. A ball is drawn at random and is found to be of blue colour. Find the probability that it is the only blue coloured ball. I am not getting a start. Like do we ...
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3answers
91 views

Three six-sided dice are rolled. Denote the greatest value rolled to be x and the smallest value rolled to be y. What is the expected value of x-y?

Three six-sided dice are rolled. Denote the greatest value rolled to be $x$ and the smallest value rolled to be $y$. What is the expected value of $x-y$? Any help is greatly appreciated. I'm not sure ...
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1answer
349 views

Square Root of a Characteristic Function.

For any $\phi$ a characteristic function, is there a characteristic function $\Phi$ such that $\Phi ^2=\phi$? I know that if $\phi$ is a characteristic function, then $\phi ^2$ is as well. Also that ...
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1answer
500 views

What is the probability that if six letters are selected from the alphabet with replacement that the chosen letters can be used to spell ASSESS?

In an arrangement of 6 letters where each letter is selected at random from the alphabet (letters can be selected repeatedly). What is the probability of getting the word 'ASSESS' if after selecting ...