ThinkDifferent

total no of ways= 16!/(8!)^2 = 12870.

no of ways in which the path goes through the abysm are:
= (8!/(7!*1!))^2 + (8!/(6!*2!))^2 + (8!/(4!)^2)^2 + (8!/(2!*6!))^2
= 64 + 784 + 4900 + 784
= 6532.

thus the required probablity is:
= 6532/12870
= 0.5075
 

That's wrong!

I guess, you forgot that, once the man goes into the gutter, he can never reach the END. There is no question of "through the abysm"! [:)]
 

Total Number of ways to reach the End = 16!/(8! * 8!) = 12870

From this we need to remove all those paths that pass through the 4 abysms.
1.First Abysm [1 RIGHT, 7 UP]
Total number of paths passing through this one = 8C1 * 8C1 =64

2.Second Abysm [2 RIGHT, 6 UP]
Total number of paths passing through this one = 8C2 * 8C2 =784

3.Thrid Abysm [4 RIGHT, 4 UP]
Total number of paths passing through this one = 8C4 * 8C4 =4900

4.Fourth Abysm [6 RIGHT, 2 UP]
Total number of paths passing through this one = 8C2 * 8C2 =794

So total number of paths to be removed = 64+784+4900+784 = 6532


Required Prob =(12870-6532)/12870
= 0.49246309246309246309246309246309
 

1- {(8C1 + 8C2 + 8C4 + 8C2) / 16C8}
= 1- 134 / 16C8
 

When 4 absyms are put in grid of 8*8 in the diagonal, the probability of reaching the End becomes 0.5.
 

I do not understand, "Why do you consider 16 steps?"

The fact is, after 8 steps only, you know whether he will reach the End or not! As simple as that.

I guess now you know what the correct answer is!

-Sujeet