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# Dividing 200 people in 3 unequal rooms and probability of certain combinations

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We have 3 rooms (say 1, 2 and 3). Room 1 can take 50 people, room 2 can take 50 and room 3 can take 100. I understand that the number of ways of assigning 200 people to these 3 rooms is $$200 choose 50 150 choose 50$$. The question is what is the probability that Alice (A) and Bob (B) end up in the same room while Charlotte (C) ends up in a different room.

My solution

I first divided the event E = {A and B in same room, C in different room} in the disjoint events E1 = {A and B in room 1, C in room 2} $$cup$$ {A and B in room 1, C in room 3} , E2 = {A and B in room 2, C in room 1} $$cup$$ {A and B in room 2, C in room 3} and E3 = {A and B in room 3, C in room 1} $$cup$$ {A and B in room 3, C in room 2}.

I calculated the possible cases for E1 in the following way: $$197 choose 48149 choose 49$$
$$+$$ $$197 choose 47$$ $$149 choose 50$$. My reasoning for the case A and B in 1 while C in 2 was that I first pick the people that will make up room 1 alongside A and B (excluding C) and then pick the people that will make up room 2 alongside C (room 3 will then be uniquely determined). The reasoning for the case A and B in 1 while C in 3 was analogous.

Similar reasonings were employed to calculate the favorable cases for E2 and E3:

$$|E2|$$ = $$197 choose 49148 choose 48$$
$$+$$ $$197 choose 50$$ $$147 choose 48$$

$$|E3|$$ = $$197 choose 49148 choose 50$$
$$+$$ $$197 choose 50$$ $$147 choose 49$$

The probability was then simply calculated by summing the favorable cases for E1, E2 and E3 and finally dividing by the total number of cases.

I am wondering if this method is correct. Furthermore, if it is correct, I am wondering if there is a simpler method since this method requires dealing with awfully big numbers.

## Deriving the log likelihood of the observed data

Hi, I am deriving the log likelihood of the observed data in part a. However, I am not sure if I am deriving it the right way.

Below is my solution:

Yi ~ N(mu, sigma^2) = f(Yi = xi) = ∅ (xi; mu, sigma^2) (for ri =1)

Yi ~ N(mu, sigma^2) = f(Yi = xi) = Φ (xi; mu, sigma^2) (for ri =0)

Then I multiply these two terms

f(xi,ri) = ∅ (xi; mu, sigma^2)^ri*Φ (xi; mu, sigma^2)^1-ri

and then I form the likelihood equation:

L(mu,sigma^2|x,r) = n i=1 prod ∅ (xi; mu, sigma^2)^ri*Φ (xi; mu, sigma^2)^1-ri

Is this the correct way to derive the equation in part a?
I have also attached the picture of the question.

## How would you authenticate requests for player specific data in a server authoritative game?

I am creating a server authoritative game that uses a public game service API to manage player specific data in the game. The client will authenticate with the auth server, get a token, then use that token to access player specific data in the API. This works well if the API calls to the game services occur on the game client. However, in a server authoritative model, the game server itself would need to make those calls on behalf of the client. My current idea is to replicate the access token to the server and then have the server figure out which token to use when it is requesting data for a specific client. I was curious if this is the best way to do this or is there another approach I should be considering?

## TypeError: ‘NoneType’ object is not iterable using Zonal Statistics

I’m trying do looping zonal statistics to many polygons in a folder, and rasters in aonther folder. Each polygon from a specific folder need to get the mean values from a many rasters from another folders, like slope, standard deviation, etc. But when I run this code the error is:

Traceback (most recent call last):
File “D:/ZONAL1.py”, line 10, in
for shp in shapefiles:
TypeError: ‘NoneType’ object is not iterable

What this means?
My code:

``````import arcpy
import os

arcpy.env.workspace = r'D:pontos_modelagembuffers'
arcpy.env.workspace = r'D:slopes'
arcpy.env.overwriteOutput = True
output = "D:tractebelpontos_modelagembufferdbf"
rasters = arcpy.ListRasters()
shapefiles = arcpy.ListFeatureClasses()
for shp in shapefiles:
for rst in rasters:
table_out = output + "\" + shp + "_Zonal_Est"
arcpy.gp.ZonalStatisticsAsTable_sa(shp,'FID',rst, table_out,"MEAN")
``````

## Solve \$1= (x^2+y^2)^{1/2} +((x-1)^2+y^2)^{1/2}.\$

Solve $$1= (x^2+y^2)^{1/2} +((x-1)^2+y^2)^{1/2}$$
for real numbers $$x,y$$. A accept a curve if it is possible or a picture. It may be a cusp.

## QGIS – after restarting project WFS Layer only displays after opening attribute table and changing zoom

When i close my project and restart, some WFS Layers are behaving strange. They only display the data after loading/opening attribute table and changing zoom. I’ve tried refreshing with F5 but it is not helping. Any ideas?

## Which is the real Euler’s Rotation theorem?

There are two quite different, but useful, theorems related to 3D rotation that are referred to as “Euler’s Rotation Theorem“.

The first definition is given as a quotation in the book by Kuipers

• Any two independent orthonormal coordinate frames can be related by a sequence of rotations (not more than three) about coordinate axes, where no two successive rotations may be about the same axis.

There is a similar, but more pithy definition, from Wolfram MathWorld

• An arbitrary rotation may be described by only three parameters.

This is related to the three-angle representation of an arbitrary rotation by Euler, roll-pitch-yaw, Tati-Bryan angles etc.

The second definition is from Wikipedia

• When a sphere is moved around its centre it is always possible to find a diameter whose direction in the displaced position is the same as in the initial position. (Also given in latin)

This related to the angle-axis representation of an arbitrary rotation. This version of the Rotation Theorem is found in many other places as well, including here on Math Overflow.

Both definitions are related to rotation, and the second at least has a reference to Euler 1776. The first is related to Euler axes and Euler angles, seems like the sort of thing Euler might have figured out, but Kuipers does not provide a reference.

Euler was notorious for not publishing much. Somebody, post Euler, has declared these theorems as Euler’s Rotation Theorem. Which one is it? Is there a more nuanced way to refer to them?

## Using Overpass to find intersections with 3 or more links

I’m trying to find all the intersections (nodes) with 3 or more links (ways) in an area as an indicator for walkability. Complete beginner to the Overpass API and this has me pretty stumped. OSM clearly shows which ways are part of a node, suggesting that this should be very doable. Can’t be much more specific as I don’t even know where to start, so thanks in advance for any help!

## USN-5931-1 Why is this security patch only for ESM/ Ubuntu Pro

18.04 LTS still has general support until April 2023, the patch for USN-5931-1 was released March 7th 2023. As march is before April, and this is a security patch, this should have been included in support offered via LTS. Why was this decided to be only offered for ESM?

## A riddle involving the axiom of choice

So i found a riddle on reddit which is a consequence of the axiom of choice.

There is a house with 100 rooms, and each room contains countably many boxes indexed with the natural numbers. Each box contains a random real number, which is the same over all the rooms (that is, box n contains the same real number in every room).

100 set theorists play a game. Each person will go into a unique room and open as many boxes as they like (perhaps countably many) as long as they leave at least one box in their room unopened. Then, each of them need to pick an unopened box in their room, and guess what real number is inside of it.

In order to win, 99 of them need to guess correctly.

The mathematicians can discuss a strategy beforehand, but after they go into their respective rooms, no more communication is allowed. What is a 100% winning strategy for this seemingly impossible task?

My question is why can the mathematicians agree on a specific representative for each equivalence class? I thought that the axiom of choice guaranteed a choice function existed, but didnt specify it. But for the solution to the riddle, they would need to know what the choice function is. Can someone explain?

## problem with Ubuntu Pro blocked my pc

I am user of Bodhi 18.04 LTS Bionic 32 bit wich based on Ubuntu.I update to nvidia-304 and now i can’t enter on my Bodhi os. because it’s show me Ubuntu Pro at the beginning.After insert my password and Token, he show me the error:

WARNING:root:failed to process /proc/version_signature.

Anyone have solution to help me ?

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