Difference between revisions of "2024 SSMO Speed Round Problems/Problem 6"
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| + | Let <math>a_n</math> equal <math>1</math> if the nth house is occupied and <math>0</math> if the nth house is unoccupied. Thus, we are trying to find <math>\mathbb{E}[a_1+a_2+a_3+a_4] = 4\mathbb{E}[a_1]</math> by the Linearity of Expectation. | ||
Revision as of 19:38, 2 May 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.
