A grid of 4x4 is given

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A ball is located at the center of the grid which is to perform a 5 step random walk with equal probability in any direction.

If it touches the grid's edges it'll stop

I need to find the probability of the event in which the ball stays in the inner 3x3 grid (that's to say, it never touches the grid's edges)

---What I've already done:

I was told to use the multinomial distribution, however I found that some cases doesn't even exist (e.g: When I tried to find the probability (using a multinomial distribution) in which the ball moves 2 steps down and 3 steps up, I realized that it was impossible because it cannot give the three up steps first), also I realized that the ball can never touch the corners

Instead of that I did a kind of tree to help me to work out how many 5 step walks were posible in which the ball remained at the inner 3x3 grid. I found out that that number is 1296

But in order to work out how many walks are posible in which the ball stays at the grid's edges, it must be considered that there are from 2 step to 5 step walks

I mean, is there any "easier" way of working out this rather than doing an exhaustive analysis?

.__.__.__.__.

| | | | |

.__.__.__.__.

| | | | |

.__.__o__.__.

| | | | |

.__.__.__.__.

| | | | |

.__.__.__.__.

A ball is located at the center of the grid which is to perform a 5 step random walk with equal probability in any direction.

If it touches the grid's edges it'll stop

I need to find the probability of the event in which the ball stays in the inner 3x3 grid (that's to say, it never touches the grid's edges)

---What I've already done:

I was told to use the multinomial distribution, however I found that some cases doesn't even exist (e.g: When I tried to find the probability (using a multinomial distribution) in which the ball moves 2 steps down and 3 steps up, I realized that it was impossible because it cannot give the three up steps first), also I realized that the ball can never touch the corners

Instead of that I did a kind of tree to help me to work out how many 5 step walks were posible in which the ball remained at the inner 3x3 grid. I found out that that number is 1296

But in order to work out how many walks are posible in which the ball stays at the grid's edges, it must be considered that there are from 2 step to 5 step walks

I mean, is there any "easier" way of working out this rather than doing an exhaustive analysis?

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