in Probability
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in Probability
518 views

2 Comments

A answer
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@arvin

Dont have key.

Please explain your approach.
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2 Answers

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Best answer

it can be done in different ways but i am using here hit and trial as that can be lengthy....

note: non decreasing(for a0,a1) means a0<=a1

consider it for single symbol : <a0>

non decreasing sequence= (0), (1),(2) = 3sequence

total sequence possible = 3( ao can contain 0,1,2)

probability(1symbol) = 3/3 =1 [email protected]

now consider it for two symbol sequence : <a0,a1>

so non decreasing sequence will be : (0,0) , (0,1) , (0,2) ,(1,1) ,(1,2) ,(2,2) =6

total sequences from 3 symbols = 3*3 (a0 can take 3values = 0,1,2 also a1)

probability(2 symbols)= 6/9 =2/3...[email protected] 

similarly for three symbols <a0,a1,a2>

non decreasing sequence= (0,0,0), (0,0,1), (0,0,2), (0,1,1), (0,1,2), (0,2,2) ,(1,1,1), (1,1,2) ,(1,2,2), (2,2,2)=10sequences

total sequences = 3*3*3 

probability(3 symbols) = 10/27 [email protected]

checking option A for above case :

(n=1) = (1+3*1 +2)/(2 *3) = 6/6 =1 [email protected] proved

(n=2) = (2*2+ 3*2 +2) / (2*3*3) = 12/18 = 2/[email protected] proved

(n=3) = (3*3 +3*3 +2)/(2*3*3*3)=20/54=10/[email protected] proved.

so answer is option A.

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4 Comments

@MiNiPanda

I ll try that but solving equations, especially non homogeneous is a bit booring task :p

Help me with denominator part from options
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@minipanda the equation will become : x3-2x2+x-1 =0

but how to solve this..

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@arvin

Actually , for homogeneous part ,equation becomes "r^2 -2r+1".

Find its roots, then find solution for homogeneous part.

Then solve for non homogeneous part.

A bit lengthy process.

https://www.tutorialspoint.com/discrete_mathematics/discrete_mathematics_recurrence_relation.htm

This will help or there are YouTube vid.

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Answer is A.

Lets Tn be total no of non decreasing sequence possible.

and 3n be the total no. of sequence possible.

Hence probability be Tn / 3n .

Tn can be calculated in following ways