Question

Use the reflection principle to find the number of paths for a simple random walk from S0=2 to S15= 5 that do not hit the level at k=6.

Answer 1

Answer:-

Given That:-

the number of paths for a simple random walk from =2 to = 5 that do not hit the level at k=6.we are using the reflection principle

Now we have to find the number of paths for a simple random work form. Given values .

By using the reflection principle

do not hit k = 6

Now we can show by graph:

From we can not any points

Here Either Hits k=6

(or) not

Consider Type 1 and Type 2.

Here

paths that hit k = 6 is Type (1)

paths that do not hit k = 6 is Type(2)

using Reflection principle at any path that

hit k = 6 and seen as path From to

Then total 2 to 5 Insteps

Total 2 to 5 = Type 1 + Type 2

Total 2 to 5 can be write as 3 ups + 6 ups +6 downs.

Type 1 can be written as 2 to 1 in 15 steps.

= 10 ups , 5 downs

That which

Total 2 to 5 Insteps = Type 1 + type 2