Difference between revisions of "2024 SSMO Speed Round Problems/Problem 6"
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==Problem== | ==Problem== | ||
| − | There are <math>4</math> people and <math>4</math> houses. Each person independently randomly chooses a house to live in. The expected number of inhabited houses can be expressed as <math>\ | + | There are <math>4</math> people and <math>4</math> houses. Each person independently randomly chooses a house to live in. The expected number of inhabited houses can be expressed as <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. |
==Solution== | ==Solution== | ||
Revision as of 14:18, 10 September 2025
Problem
There are 
people and houses. Each person independently randomly chooses a house to live in. The expected number of inhabited houses can be expressed as 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
Let 
equal 
if the nth house is occupied and 
if the nth house is unoccupied. Thus, we are trying to find ![$\mathbb{E}[a_1+a_2+a_3+a_4] = 4\mathbb{E}[a_1]$](http://latex.artofproblemsolving.com/2/8/b/28b16f035372bc7b85c7c31849f99abdb0d0c0ee.png)
by the Linearity of Expectation. Instead of calculating the probability that a single house is occupied, we instead calculate the probability that it is unoccupied. Each person has a 
chance of residing in that house, and thus a 
chance of not residing in that house. The probability that none of the people stay in that house is 
. Thus, the probability that any given house is occupied, or ![$\mathbb{E}[a_1]$](http://latex.artofproblemsolving.com/4/b/d/4bd93b9d2ffbcd175bbb28d9a3b26778ff136e06.png)
, is 
. Thus, our answer is 
, 
.
-Vivdax
