1
vote
0answers
47 views

How to convert a problem to a stars and bars problem?

Continued question from here. How can I convert a problem like: $$\def\x{x_}\x1+\x2+\x3+\dots =15$$ with $\x1\leq3$ Back to a simple stars and bars problem such as $$y_1+\x2+\x3+\dots=33$$ How ...
0
votes
1answer
32 views

probability rolling a dice 5 times

I can't solve this problem: What is the probability that, when rolling a dice 5 times, the number of times when you get a 1 or 2 is greater than the number of times when you get a 6. any help?
1
vote
0answers
35 views

Expected frequency of most frequent die roll

Suppose we have an fair $m$-sided die, and we roll it $n$ times. What is the expected frequency $E(n, m)$ of the most frequently rolled face? If we fix $n$ we can calculate $E(n,m)$ like so. Let ...
2
votes
2answers
32 views

Collision of 8 Digit, Base-36 Numbers

I have an algorithm that generates a random 8 digit, base 36 number with uniform distribution. Thus, this algorithm can generate $36^8$ unique numbers. I run my algorithm 10,000 times, and write ...
0
votes
4answers
30 views

Probability: Linear Seating Arrangement

Okay, I'm new at probability and statistics, so please try to answer this as thoroughly as possible and explain why you did everything, from using a specific number to why using factorials and ...
0
votes
3answers
75 views

How to use stars and bars(combinatorics)

How to use the stars and bars method? Say I want to find number of combinations I can get with $x_1+x_2+x_3+x_4=22$ Where $x_i\in\mathbb{N}$ Is this the correct time to apply the method?
2
votes
3answers
262 views

Probability of dying from smallpox?

A family of four is infected with Variola major. There is a fatality rate of 30%. Calculate the probability that... Here are my attempts, The probability that nobody dies, ...
0
votes
2answers
35 views

Boxes and colored balls with replacement

Suppose there are $n+1$ boxes numbered from $0$ to $n$. The $i$-th box contains $i$ white balls and $n-i$ black balls. A box is chosen randomly and a ball is selected from the box, after that the ...
2
votes
2answers
70 views

Pólya's urn scheme, proof using conditional probability and induction

Problem An urn contains $B$ blue balls and $R$ red balls. Suppose that one extracts successively $n$ balls at random such that when a ball is chosen, it is returned to the urn again along with $c$ ...
0
votes
1answer
27 views

Dice, balls and boxes probability problem (conditional probability)

Problem Suppose there are two boxes $A$ and $B$ such that $A=\{\text{5 red balls and 3 white balls}\}$, $B=\{\text{1 red ball and 2 white balls}\}.$ A dice is thrown, if the result is $3$ or $6$, a ...
2
votes
1answer
35 views

Colored balls in three boxes (conditional probability problem)

Problem Suppose there are three boxes numbered with twenty balls in each of them. The first box contains twenty white balls; the second, fifteen, and the third,ten; the rest of the balls are black. ...
2
votes
1answer
69 views

A possible incorrect application of Law of Large numbers

A friend left this teaser for me. He asked me to first compute: $$ \lim_{n \to \infty} \frac{\binom{2n}{n}}{2^{2n}}$$ Using Stirling's approximation (and another method), I got the answer as $0$. ...
1
vote
2answers
26 views

Ice cream combinatorics question

An ice cream shop sells ice creams in $5$ possible flavours: vanilla, chocolate, strawberry, mango and pineapple. How many combinations of $3$ scoops cone are possible? [note: repetition of flavours ...
0
votes
2answers
70 views

If bridges between islands collapse independently with probability $p$, what is the probability that islands remain connected?

This is a follow-up to Probability Question: Bridge problem. There are $n$ islands in the ocean. Each island is linked by a single bridge to each other island. The probability of each bridge ...
1
vote
1answer
43 views

Probability Question: Bridge problem

There are $n$ islands in the ocean. Each island is linked by a single bridge between each and every unique pair of islands to ensure no island is isolated from the others. The probability of each ...
0
votes
1answer
31 views

Probability formula related to distribution balls in boxes.

Problem Suppose there is a distribution of $N$ distinct balls in $n$ different boxes such that each ball has the same probability to be in any box. Let $A_i=\{\text{the i-th box is not empty}\}$. ...
2
votes
2answers
42 views

Probability problem: n different balls in n different boxes

Problem Suppose $n$ different balls are distributed in $n$ different boxes. Calculate the probability that each box is not empty when distributed the balls. I'll define the sample space as ...
0
votes
1answer
55 views

distribution of books among students

There are $p$ students and $q$ books where $q>p$ and all books are different, but each student will get a minimum of $1$ book and a maximum of $(p – 1)$ books. Find the total number of ways of ...
2
votes
2answers
37 views

Subset Probability to Element Probability

Is there any way to match (or map) from Subset Propabilities to Element Probabilities? Suppose that John may select x-sized subsets from a population of N items. In every subset he has exactly x ...
7
votes
1answer
129 views

The Day Camp Stacking Game

My friend works at a day camp as a counselor and he told me about an interesting game he plays with his group of kids. You have a perfectly shuffled, regular $52$-card deck and a group of $2 \leq n ...
3
votes
0answers
75 views

What's so special about binomial coefficients that someone decided to organize them in a triangle?

I know that binomial coefficients are related to figurate numbers (which were studied by Greeks a loooong time ago, because of its connections to geometry). I also understand how the Pascal's triangle ...
1
vote
1answer
34 views

What is the probability that each of the vehicles will be made to carry at least one local tourist?

Three vehicles (one blue, one green and one grey) with a carrying capacity of 8 passengers each are to be used to ferry 18 international tourists and 5 local tourists (who are a family) from OR Tambo ...
1
vote
1answer
23 views

What is the probability that the maximum number of shots fired successively from a type A gun is $2$?

A gun salute always takes place at the funeral of a military leader who has died in a certain country. (The $21$ gun salute where $21$ rounds are fired - is the most common for the most senior ...
0
votes
2answers
30 views

Probability that the first $2$ balls are white, given that the sample contains exactly $6$ white balls

An urn contains $30$ white and $15$ black balls. If $10$ balls are drawn without replacement, find the probability that the first $2$ balls are white, given that the sample contains exactly $6$ ...
0
votes
2answers
47 views

6 Professors and 8 floors - expected value

I have this problem I need help with. There are 6 professors on an elevator that has 8 floors/stops. Each professors exits the elevator randomly(1/8 chance). What is the expected value E(X) of stops ...
1
vote
4answers
74 views

Probability of drawing at least 1 red, 1 blue, 1 green, 1 white, 1 black, and 1 grey when drawing 8 balls from a pool of 30?

Given a pool of 30 balls (5 of each color). When drawing 8 balls without replacement, what is the probability of getting at least one of each color? Related: Probability of drawing at least one red ...
1
vote
2answers
90 views

Probability of assigning balls into buckets, where each bucket has a certain capacity.

I'll start with a specific example of what I am trying to solve: I have eight balls to be randomly placed into four buckets. Buckets #1-3 have the capacity of 2, 2, 3 respectively, while bucket #4 ...
8
votes
0answers
136 views
+100

How to estimate $Pr[vr_i=ur_i]$ in the presence of rotations

Suppose we want to compute the probability that for two different random vectors (with elements that are $0$ or $1$), denoted by $v$ and $u$, multiplying them with the rotations of a random vector $r$ ...
-1
votes
1answer
30 views

In how many of the possible arrangements will both end balls be of the same colour? [closed]

6 blue balls, 4 red balls, and 2 white balls are placed in a straight line. In how many of the possible arrangements will both end balls be of the same colour?
0
votes
1answer
31 views

Probability for incomplete information

Let's say there are 10 teams: A-J. Only 1 team wins, others lose. Probability of any team to win is unknown (different for each team) and to be calculated. Not all teams participate in each game. ...
0
votes
1answer
34 views

How to calculate probability that a team will win

Let's say there are 10 teams: A-J. Each team always participate in each of the game. Only 1 team wins, others lose. Probability of any team to win is unknown (different for each team) and to be ...
1
vote
1answer
35 views

Rolling 6 dice and 3 on the same side

I found similar answer for 'Rolling 6 dice and 4 on the same side'. Will exactly the same rule apply for 3 dice? Instead of 15000 (according to the above link), I found somewhere 14700. $${6 ...
0
votes
3answers
21 views

What is the probability that the newest of the first edition is next to the oldest of the second edition?

There are six copies of a certain textbook in a school library. They were all purchased at different times. 3 are of the first edition and the other 3 are second edition. They have been returned ...
0
votes
0answers
20 views

Bounding a specific function of binomial coefficients

While trying to directly prove the existence of expander graphs (e.g. http://www.cs.toronto.edu/~avner/teaching/S6-2414/TUT2.pdf), one uses the following inequality: $$\sum_{s=1}^{n/2} ...
-2
votes
2answers
59 views

What is the probability that no members of a married couple sit next to each other [closed]

Five people (A, B, C, D and E) dine at a round table. They take their places at random. A is married to B. C is married to D. E is single. What is the probability that no members of a married ...
0
votes
1answer
41 views

Round table seating - expected value

There are $p$ women, $s$ men and $p+s$ seats in a round table. Let $X$ be the number of women who sit between two men. Find the expected value of $X$. I know that expected value of $X$ is given by ...
3
votes
4answers
268 views

2 of 3 dice are selected randomly and thrown. What is the probability that one of the dice shows 6

1 red die with faces labelled 1, 2, 3, 4, 5, 6. 2 green dice labelled 0, 0, 1, 1, 2, 2. Answer: 1/9 Please can you show me how to get the answer. I'm confused about joining the events of choosing 2 ...
0
votes
1answer
56 views

2011 AIME Problem 12 — Probability: 9 delegates around a round table

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one ...
0
votes
1answer
23 views

Bernouilli trial with variable number of experiments

I'm kinda stuck on a probability problem I encountered in designing a game. Here is its description : I'm calculating the number of turns (Tf) before a integer variable (A) reaches 0. Each turn, A ...
0
votes
1answer
22 views

What is the probability of getting at-least one even digit for 5 trials.

Let we have through a fair dice. What is the probability of getting at-least one even digit for 5 trials. In this case I have evaluated the number of trials as follows: The probability ...
2
votes
2answers
53 views

A coin is tossed six times. What is the probability of getting at least four heads on the tosses?

A coin is tossed six times. What is the probability of getting at least four heads on the tosses? I have solved the problem like this: probability of getting 2 tail = ${}_6C_4 \times ...
0
votes
0answers
14 views

Slots Machine Matching feature 2

I'm designing a slot machine. I need to find the number of combinations that two matching icons will appear side-by-side in a 3X5 window (3 rows, 5 reels (columns)) A match gives the user some ...
1
vote
1answer
52 views

How to compute the expected number of unwatched rooms?

Imagine that I have a number of rooms, r, that I want to have watched. So I install one video camera in each room and have televisions that can show what's going ...
3
votes
5answers
1k views

Probability of guessing a PIN-code

A friend and I recently talked about this problem: Say my friend feels a little adventurous and tells me that exactly three of four digits of his PIN-code are the same, what is the probability that I ...
1
vote
0answers
40 views

A variation of the menage problem

A combinatorics problem I am chewing on without success is: 3 couples and 40 others are to be arranged randomly in a row. What is the probability that no two couples sit together ? I have looked at ...
1
vote
1answer
35 views

A question involving the throw of seven dice, which of my answers is correct?

Seven dice are thrown, what is the probability that all numbers show up on the dice? My first answer uses the logic that if all numbers show up and you throw seven dice, then one number is repeated ...
2
votes
2answers
41 views

Selecting 180 days from 366: the probability of even distribution across months, or not having September among the first 30

In a draft lottery containing the 366 days of the year (including February 29). Select 180 days (draw 180 without replacement). a) What is the probability that the 180 days drawn are evenly ...
5
votes
4answers
135 views

Probability that a word contains at least 3 same consecutive letters?

Assume we have a word of length $n$ and an alphabet of length $26$ (the small letters a through z, if you want so. How likely is it that this word contains at least $k := 3$ consecutive letters of ...
0
votes
3answers
82 views

Birthday paradox, huge numbers

Pick x random "birthdays", say $10^9$. What are the chance of a collision, given $2^{160}$ possible "days"? I'm trying to estimate the collision rate of sha1 hashes, but the calculation is too big ...
3
votes
3answers
45 views

Probability of having 4 aces after taking turns to pick cards

I've started to learn probability and I get stuck with the following problem: My friend and I are playing a card game with 36 unique cards. There are four suits (diamonds, heart, clubs and spades), ...