0
votes
2answers
25 views

Box containing Red and black balls

A box containing 4 red balls and 6 black balls . Three balls are selected randomly from the box one after the another, without replacement . The probality that the selected set has one red ball and ...
1
vote
1answer
22 views

Number of ways of distributing 7 distinct balls into 7 distinct boxes with exactly one box with 3 balls.

This is what I tried. I can distribute the balls in four ways: 1) 3,1,1,1,1 2) 3,2,1,1 3) 3,2,2 4) 3,4 For 1) I can first pick 5 boxes in ${7 \choose 5}$ ways and then pick 3 balls in ${7 \choose ...
0
votes
0answers
38 views

How many Possible Combinations exist?

I have $120$ coins and $21$ buckets. Each bucket can hold $0$ to $20$ coins. How many possible coin/bucket combinations are there?
0
votes
1answer
42 views

Calculating partitions using multiplication rule

A teacher wants to create 4 study groups of 5 students from a class of 20 students. How many ways are there to do this? The answer is $\binom{20}{5}\binom{15}{5}\binom{10}{5}\binom{5}{5}$. I don't ...
2
votes
1answer
158 views

Probability that the second-best player finishes second in a single-elimination tournament, given that better players always defeat weaker players?

A chess tournament (single-elimination format) has 16 players. Suppose that no two players have the same strength, and that each player always defeats the players weaker than himself/herself (i.e. no ...
0
votes
1answer
19 views

Probability that a company is worth $xM after y years, if its value can only stay the same or double every year?

Let's say a company is worth \$1M. Each year, the value of the company eithers stays the same with probability $\frac{1}{2}$, or doubles with probability $\frac{1}{2}$. What is the probability that ...
1
vote
4answers
70 views

Probability of at least 3 red balls given 4 choices in a bag of 4 red balls and 4 black balls?

Let's say there are 8 balls in a bag, where 4 are red and 4 are black. If I choose four balls from the bag without replacement, what is the probability that I will choose at least 3 red balls? My ...
1
vote
2answers
30 views

Probability of being selected twice in a week given a set of n people?

Let's say a child is selected out of a group of 10 students each day to stay after school and help clean the classroom. What is the probability that a particular child is selected exactly twice during ...
1
vote
2answers
19 views

K Zeros between 14641

One Writes K- Zeros between every two digits of the number 14641. What is the square root of the number obtained? I want to know if there is a better way of writing out the solution. As of now I know ...
0
votes
1answer
25 views

Find probability of 4th smallest number?

Seven numbers are selected from the numbers (1, 2, 4, 8, 9, 10, 11, 15, 17) without replacement. What is the probability that the 4th smallest number is 9? I'm not sure if I'm getting the correct ...
0
votes
4answers
42 views

How many 3 digit even numbers can be formed using (0, 1, 2, 3, 4) and no repetition?

My solution to the problem is as follows: The answer I get is 27. My reasoning is that the last digit must be even, so for that position there are 3 choose 1 possibilities. Then the first digit ...
1
vote
2answers
63 views

Grid problem - combinatorics? [on hold]

In a city, streets are laid out as a grid. Your home is at location (0,0) and work is at location (10,10). You walk to work taking only steps up and to the right. (a) How many distinct ways can you ...
1
vote
1answer
20 views

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 ...
0
votes
1answer
36 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: ...
0
votes
1answer
15 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 ...
0
votes
0answers
39 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 ...
0
votes
1answer
30 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 ...
3
votes
1answer
75 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 ...
1
vote
1answer
62 views

Combinatorics of a tournament where one wins by taking either three games in a row or four in total

Two teams play each other repeatedly until either one of them wins three games in a row or one of them wins a total of four games. What are all the ways in which the tournament can be played? What ...
-1
votes
3answers
171 views

Probabilities related to the sum of four dice

Suppose we have 4 fair six-sided dice of different colours and faces numbered 1,2,...,6 are rolled independently. (a) How many ways can a total of i. 4 ii. 5 iii. 6 be obtained? (b) Compute (to ...
3
votes
1answer
57 views

Combinatorial Analysis: Fermat's Combinatorial Identity

I was looking through practice questions and need some guidance/assistance in Fermat's combinatorial identity. I read through this on the stack exchange, but the question was modified in the latest ...
1
vote
1answer
40 views

How can I calculate $ \sum_{j=0}^{49}\binom{100}{2j+1}p^{100-(2j+1)} q^{(2j+1)} $?

I got the following formula when I tried an exercise in probability: $$ \sum_{j=0}^{49}\binom{100}{2j+1}p^{100-(2j+1)} q^{(2j+1)} $$ where $p+q=1$. These are the "odd" terms in the expansion of ...
2
votes
1answer
45 views

How can I simplify $ \sum_{r=0}^{m-1}r^3\frac{\binom{m}{r}(m-r)!\begin{Bmatrix} n\\ m-r \end{Bmatrix}}{m^n}$?

Let $N$ and $M$ be sets with $n$ and $m$ elements respectively with $n>m$. Randomly assign a function $f:N\to M$. Suppose that the probability of each element in $N$ being assigned to any ...
1
vote
1answer
26 views

raBinomial distribution with dependent trials?

I need your help with following problem: String with n characters is given. For each character in string there is probability p that it is wrong. Now you take a sliding window of length k, k<= n, ...
0
votes
3answers
53 views

Need help with flaws in statistical reasoning

The problem is as follows - there are three couples and six chairs in a row. The six individuals are seated at random. What is the chance that at least one couple will be seated together? Here's my ...
0
votes
2answers
43 views

Probability of each outcome from dice notation

In the "dice notation", XdY means you rolls X number of Y-sided dices, and adds the results together to get the final outcome. For example, on 3d3 distribution, you can get number from 3 to 9, and ...
1
vote
4answers
33 views

dealing cards probability

If a standard deck of cards is deal to 4 players, 13 cards each, how many possibilities are there assuming that it matters which player gets but card order does not matter. Why is the answer not (52 ...
0
votes
1answer
29 views

Rolling a three sided die six times?

Consider the problem of a rolling a three-sided die six times (independently). The probability of seeing 1 is 0.5, 2 is 0.25, and 3 is 0.25. With this model, I have been given the claim that: We ...
0
votes
2answers
23 views

combinatorics question sampling without replacement

Suppose a bag has $x$ blue marbles and $y$ red marbles, and the marbles are picked one at a time without replacement. Would the probability that all blue marbles are picked before red marbles be ...
1
vote
2answers
34 views

How many different DNA sequences of length 4 consist of exactly two different letters?

Note: $P(A)=p_A, P(C)=p_C, P(G)=p_G, P(T)=p_T$ My attempt: I tried to make a list of every single possible sequence. How can I solve this question more efficiently?
0
votes
4answers
89 views

Random walk on a finite square grid: probability of given position after 15 or 3600 moves

An ant is walking on the squares of a 5x5 grid - it starts in the center square. Each second, it will choose (with equal probability) to do one of the following: Move north one square Move south ...
3
votes
2answers
47 views

Expected number of output letters to get desired word

I am using a letter set of four letters, say {A,B,C,D}, which is used to output a random string of letters. I want to calculate the expected output length until the word ABCD is obtained; that is, the ...
2
votes
2answers
103 views

Hat Matching Problem Expectation

I have an interesting problem in the context of the hat matching problem: There are n people with hats at a party. Each person randomly grabs a hat. A match occurs if a person gets his own hat. I'd ...
3
votes
1answer
42 views

Applying Combinatorics to College Applications

From a list of 4,000 colleges, I have a list of 50 colleges. From that list of 50 colleges, I now plan to choose 15 colleges to apply to. Each of the 50 colleges has a scale value (how well the ...
1
vote
1answer
27 views

Independent, Probabilistic, Event occurs once in a set of 10, Has 10 orientations. Probability?

I have ordered sets. Each index in the set is random and independent. Here is a sample of possible sets: X1 < X2 < X3 < X4 < X5 < X6 < X7 < X8 < X9 < X10: ...
0
votes
1answer
26 views

What are the odds that every team that lost the prior week would be facing a team that won the prior week?

I apologize in advance for a potentially elementary question but I cannot figure out how to even begin with this. We have 10 teams Half lost the first week, the other half won Second week, we all ...
1
vote
1answer
90 views

Limited Edition - How many C(6, 3) do I need?

A limited edition of products came out, with sets of 3 items pulled randomly from the series of 6 and sold in 3-packs. I know that C(6, 3) = 20 different possible outcomes, but I'm having trouble ...
2
votes
3answers
138 views

Odds for randomly assigning a men-only group in a team working assignment

We are partitioning a group of $30$ people in $5$ groups of $6$ persons each. We have $13$ women and $17$ men in those $30$ people and randomly drawing those people gave us a men-only group. What are ...
1
vote
1answer
65 views
+50

counting occurence of subgraphs by counting their occurence in larger subgraphs

I have a mental block in fully understanding the following notion. Let $G$ be a graph of order $n$ and $H$ a fixed small graph of order $k \le n$. Suppose that there are $d$ copies of $H$ as an ...
2
votes
1answer
74 views

Probability of a substring occurring in a string

Consider a random string of length $n<\infty$ where each digit can be between 0-9 with equal probability and a substring of length $k<n$ consisting of only zeros. What is the probability of ...
1
vote
0answers
56 views

Efficient calculation of minimal expected number of inversions

Problem: I have an array of size n with Z inversions initially and I am allowed to perform K operations where each operation can be decrease the number of inversions by 1. make a random shuffle of ...
1
vote
1answer
35 views

Combinatorics with simple substitution ciphers question

Help me out with this combinatorics question - I understand that there's 26! possible ciphers - would fixing one result in 25! ciphers? Unsure how to handle "at least" Any help is greatly ...
5
votes
7answers
3k views

$6$ women and $4$ men wait in line. If their order in line is random, find the probability that all of the women are adjacent to one another.

My thoughts on the problem are that the number of ways the women can be adjacent to each other is $5!$ and the total number of arrangements for all the people is $10!$. Is this correct?
0
votes
1answer
34 views

Count of possible hands math + Python vs Wolfram Alpha

So I wanted to write a Python program to calculate some probabilities using hyper geometric distributions. However, I seem to get probabilities over 1 sometimes, so there must be something wrong with ...
2
votes
2answers
53 views

Unordered sampling with replacement

I have difficulties with understanding of how do we count this. Why can't we just divide number of ways of ordered sample with replacement $n^k$ by $k!$, i.e. to get rid of permutations since we ...
0
votes
1answer
8 views

Subset Probability to Element Probability (part II)

Asking in conjuction with the previous question: Subset Probability to Element Probability If John selects any sized-subset (from 1 element to N elements), which is the probability of selecting ...
1
vote
1answer
33 views

Combination question involving apples and oranges

Suppose that I have n apples and n oranges, I want to arrange the apples and oranges in a row such that no apples will be next to another apple and no oranges will be next to another orange. Please ...
0
votes
0answers
25 views

Probability for a family of submultisets?

I have a multiset, e.g. something like A=[a,a,a,b,b,c] and I want to know the probability/frequency of the family of submultiset B=[x,x,o] where the x's means equal elements and the o means a ...
5
votes
0answers
117 views

Probability that at least one of four hands missing at least one suit

Deal each of four players a 13-card hand at random. What is the probability that at least one of the four hands is missing at least one suit? Let $A_i$ mean that player $i$ is missing at least one ...
0
votes
2answers
27 views

Probability and combinations intuition

This was a question randomly given in my class by my teacher: In a class of 10, each student has the same chance of getting a scholarship. Find the probability that exactly 3 of the students will get ...