Linked Questions

25
votes
10answers
12k views

What is the probability of the sum of four dice being 22?

Question Four fair six-sided dice are rolled. The probability that the sum of the results being $22$ is $$\frac{X}{1296}.$$ What is the value of $X$? My Approach I simplified it to the equation ...
13
votes
6answers
1k views

Book on combinatorial identities

Do you know any good book that deals extensively with identities obtained using combinatorial and/or probabilistic arguments (e.g., by solving the same combinatorial or probability problem in two ...
4
votes
6answers
2k views

Number of occurrences of k consecutive 1's in a binary string of length n (containing only 1's and 0's)

Say a sequence $\{X_1, X_2,\ldots ,X_n\}$ is given, where $X_p$ is either one or zero ($0 < p < n$). How can I determine the number of strings, which do contain at least one occurrence of ...
6
votes
4answers
2k views

Probability event with $70$% success rate occurs three consecutive times for sample size $n$

It has been a long time since I've done probability so I am not sure which to do (if either are correct). Thank you for taking the time to look at my work. Probability an event occurs is $70$%. I'm ...
6
votes
3answers
651 views

Number of solutions of $x_1 + x_2 + x_3 + x_4 = 14$ such that $x_i \le 6$

Let $x_1, x_2, x_3, x_4$ be nonnegative integers. (a) Find the number of solutions to the following equation: $$ x_1 + x_2 + x_3 + x_4 = 14 $$ I got $17 \choose 3$ for this. Is ...
5
votes
4answers
346 views

Number of q-ary strings of length m which do not contain k consecutive zeros

A finite q-ary-alphabet is given $$A_q = {0,1,2,...,q-1}.$$ Now I am considering the set of all finite strings over the alphabet $A_q$. I am interested on the number $$N(m,k)_{A_q}$$ of strings of ...
3
votes
3answers
3k views

Efficient algorithm to find all unique combinations of set given duplicates

The number of combinations for a 4 choose 2 is 6. But what if I have a set in which 2 of the elements are the same? Of course I could do a unique check for each item but this is too computationally ...
2
votes
2answers
743 views

Number of solutions of $x_1+x_2+\dots+x_k=n$ with $x_i\le r$

Let $n,k,r$ be positive integers. The number of all nonnegative solutions of the Diophantine Equation $x_1+x_2+\dots+x_k=n$ is $\binom{n+k-1}{n}$. Is there a general formula for the number of ...
7
votes
3answers
472 views

Number of combinations with sets of identical items

I was trying to write a python program which takes a string and prints out all possible combinations of characters from it and in turn figure out the number of such possible combinations of a string ...
3
votes
6answers
217 views

Number of bit strings of length $n$ with no $k_1+1$ consecutive 0s and no $k_2+1$ consecutive 1s.

Just as the question asks. I am trying to calculate the number of bit strings of length $n$ with a maximum of $k_1$ consecutive $0s$ and $k_2$ consecutive 1s. Of course we assume $k_1+k_2\leq n$. I am ...
3
votes
2answers
2k views

How do you calculate the sum of combinations of 1000 dice rolls?

For two dice rolls we can calculate the number of combinations for each summed total: Rolling a 2: one chance (1&1) Rolling a 3: two chances (2&1)(1&2) Rolling a 4: three chances (3&1)...
2
votes
2answers
418 views

Probability of rolling a sum of at least 18 with 10 6-sided dice

I'm trying to work out how to do this, and I'm very stuck. My grasp of probability is shaky to begin with, and generally with probability questions, I list out cases. Due to the number of ...
1
vote
2answers
2k views

How to calculate dice roll probability with more dices? [duplicate]

I'm not math graduate, so be patient. For two dices it is easy, because of small number of possibilities, also it can be easy for three, but how is it works for ten (some formula)? How to calculate ...
3
votes
2answers
770 views

Probability to obtain more than X with 3 dice.

This is a dice problem 1) I want to calculate the probability to have more than X throwing 3 dice of 6 faces. X = addition of the result of the 3 dice. 2) This is the first step but if you can also ...
2
votes
2answers
193 views

Understanding the step-hop problem mathematically

I am working on a problem where one is given n number of steps. They can take either one, two, or three steps. How many number different possible ways are there to climb the n steps? I can solve this ...

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