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I can't seem to wrap my head around how to calculate this, nor the reasoning behind it.

Say you have 3 10-sided dice. 2 dice have 8 "winning" faces and 2 "losing" faces. So an 80% chance of winning. The third dice has 5 "winning" and 5 "losing" sides, so a 50% chance of winning.

If you roll all three of these dice at the same time:

-What is the probability of at least 1 dice winning.

-What is the probability of at least 2 dice winning.

-What is the probability of all 3 winning.

From what I've seen I can calculate the probability of all 3 winning by doing:

0.8 x 0.8 x 0.5=0.32

A 32% chance of all 3 winning at the same time.

But what about the others, I can't seem to figure it out.

The well-ordering theorem follows only if the AoC is true, which means that otherwise, there are sets with no well-ordering.

Supposedly, the Real numbers is one of such sets. Does that mean the real numbers can't be shown to have a direct bijection to any aleph number? And if that's the case, does that mean 2^Aleph_0 is bigger than any aleph number?

I was watching this video, which was describing the intution behind the second derivative. I understand how the 1-dimmensional result was found, but I am quite at a loss for how to arrive at the n-dimmensional result, where the second derivative is the laplacian (the video provides the 3-dimmensional result, i.e. $f(x,y,z)$, at 00:08:08). The specific part that I'm having trouble with is finding the average of the multivariable function so that it fits the equation stated in the video, which, even more confusingly, has a single variable in it.

• Update (2024-05-08T06:31Z): I found this document, which provides a lot more detail.

Are there any known right triangles that have integer side lengths and rational angles? If not, has it been proven that none exist?

There are certain things whose number is unknown. If we count them by threes, we have two left over; by fives, we have three left over; and by sevens, two are left over. How many things are there?

Definitions

A graph G = (V, E) is a set of nodes V and a set of undirected edges E. An undirected edge is a set of two vertices.

Two graphs G1 = (V1, E1) and G2 = (V2, E2) are isomorph if there is a bijective mapping f from V1 to V2 such that f(E1) = E2. Applying f to E1 means applying it to each node in each edge in E1.

Problem

Is there an algorithm that decides if two arbitrary graphs are isomorphic in polynomial time?