a man can move in all four directions(NORTH SOUTH EAST WEST)with equal probability 1/4. If he takes 5 steps find out the probability that he is one step away from starting point.
Answers
Given : a man can move in all four directions(NORTH SOUTH EAST WEST)with equal probability 1/4.
He Takes 5 Steps
To Find : probability that he is one step away from starting point.
Solution:
Total number of ways to take 5 steps = 4⁵ = 1024
he is one step away from starting point.
2 North . 2 South ( 1 in any direction)
Hence
3 North 2 South , = ⁵C₃ = 20
2 North 3 South , = ⁵C₃ = 20
2 North . 2 South 1 East ,= ⁵C₂ x ³C₁ = 30
2 North . 2 South 1 West = ⁵C₂ x ³C₁ = 30
100 Ways
2 East 2 West ( 1 in any direction )
3 East 2 West , = ⁵C₃ = 20
2 East 3 West , = ⁵C₃ = 20
2 East 2 West 1 North , =⁵C₂ x ³C₁ = 30
2 East 2 West 1 South =⁵C₂ x ³C₁ = 30
100 Ways
1 North , 1 South , 1 East 1West ( 1 in any direction )
2 North , 1 South , 1 East 1 West ⁵C₂ * 3! = 60
1 North , 2 South , 1 East 1 West ⁵C₂ * 3! = 60
1 North , 1 South , 2 East 1 West ⁵C₂ * 3! = 60
1 North , 1 South , 1 East 2 West ⁵C₂ * 3! = 60
240 Ways
Total Ways = 100+ 100 +240 = 440 Ways
Probability = 440/1024
= 110/256
= 55/128
probability that he is one step away from starting point. = 55/128
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