Recent content by awkward

Manuel, I think you may be misinterpreting some mathematical notation. The notation $\binom{n}{m}$ means the number of ways to pick m objects from a set of n objects, which is $$\frac{n!}{m! (n-m)!}$$ I think if you work out your example of 8 balls and 5 colors, you will find the formula...

Hi Manuel, I'll give two solutions: one using generating functions (my favorite) and another using Inclusion / Exclusion. To generalize, let's say there are b balls and c colors. We want to count the number of ways to paint the balls using no more than 3 of the c colors, where $c \ge 3$. The...

Hi Ilijnnasil, Welcome to MHB. Let's say an integer $n \ge 4$ is "acceptable" if there are numbers $a_4, a_5, a_6, \dots , a_n$ where $a_i \in \{-1,1\}$ and $\sum_{i=4}^n a_i \ i = 0$. You have already shown that 7 is acceptable. Another acceptable number is 8, because $1 \cdot 4 + 1 \cdot...

Hi Jameson, The question of which hands should be counted as a four-card flush is, I think, ambiguous. The OP didn't say anything about poker, and a "four-card straight flush" is not a standard poker hand. So is a 2-3-4-5-6 of spades a four card straight flush, or is it a five card straight...

Hi Don and Soroban, These problems are harder than they look! It's very easy to mistakenly over-count. For example, in Soroban's solution above, suppose the four-of-a-kind is four aces. Some of the "other 9 cards" includes four twos among the 9 cards. But one of the other possibilities for...

The answers to your questions are 1) Effectively zero (less than 10^-1336) 2) Effectively one (more than 1 - 40 * 10^-1246) 3) Effectively zero (less than 10^-78136) Let's start by assigning names to some of your numbers: N = 200,000,000 K = 60,000 n = 10,000,000 Then if p(x) is the probability...

Actually, there are many methods for generating the subsets. If you google "algorithms for generating combinations" you will find many hits. Here is one: Algorithm to return all combinations of k elements from n - Stack Overflow If you really want to read about the subject in depth, see...

The infinite series approach is fine, but here is another way for the sake of variety: condition on the result of the first throw. Let's say E is the expected number of throws before throwing a 6. With probability 1/6, you will throw a 6 on the first throw. In this case the number of throws...

zweilinkehaende, I am not sure I understand the rules. Here is what I think you are saying. Please let us know if this is correct or not. Positive integers $x$ and $y$ are fixed in advance. You have two options: 1. Roll $x$ dice, then re-roll any that do not come up 5 or 6. Your score is...

You might find a helpful statistic here: Diversity index - Wikipedia, the free encyclopedia (I haven't used any of these measures myself.)

OK, as I wrote earlier, the most direct way to check that you are generating uniformly distributed points is to check that the number of points in a region on the surface of the sphere is proportional to the area of the region; that's the definition of "uniformly distributed". But since you're...

It's tricky to generate a point uniformly distributed on the surface of a sphere. In particular, generating the latitude and longitude uniformly will not work. You can find a discussion here: http://mathworld.wolfram.com/SpherePointPicking.html The definition of "uniformly distributed" is...

Hi Ackbach, Your answers look right to me. So maybe there is an error in the book.

Hi Ackbach, I think you're making this too hard. For (a), the sample space is simply all the subsets of size two taken from the collection of four buckets. We assume these are all equally likely if the divining rod is useless. For (b), how many of these subsets contain the two buckets with...

We will show that $\cos(n), n = 0, 1, 2, \dots$ is dense in [-1,1], so the sequence $\cos(n)$ cannot converge. Lemma: If $\theta$ is irrational, then the set $\{n \theta \pmod{1}: n \in \mathbb{Z}\}$ is dense in [0,1]. Proof: Let $\epsilon > 0$, and choose $k$ so that $0 < 1/k < \epsilon$...