Question 433456
<pre>

The other tutor's answer is incorrect.

I will assume that by "path" you mean that you always start from a,
then move right or down, never left or up.

[ a ][   ][   ][   ][   ][   ][   ]
[   ][   ][ b ][   ][   ][   ][   ]
[   ][   ][   ][ c ][   ][   ][ z ]


I will indicate with a right arrow &#8594; that the next move from 
that space will be to the right, and indicate with a down arrow &#8595;
that the next move will be downward. 

There are only three ways to go from a to b:

#1.
[ a&#8594;][  &#8594;][ &#8595; ][   ][   ][   ][   ]
[   ][   ][ b ][   ][   ][   ][   ]
[   ][   ][   ][ c ][   ][   ][ z ]

#2.
[ a&#8594;][ &#8595; ][   ][   ][   ][   ][   ]
[   ][  &#8594;][ b ][   ][   ][   ][   ]
[   ][   ][   ][ c ][   ][   ][ z ]

#3.
[ a&#8595;][   ][   ][   ][   ][   ][   ]
[  &#8594;][  &#8594;][ b ][   ][   ][   ][   ]
[   ][   ][   ][ c ][   ][   ][ z ]

But that's irrelevant, because you are given that your path goes
through b.  Therefore regardless of how you got from a to b, there 
are only these 5 ways to get from b to z, 2 go through c and 3
don't. 

1.
[ a ][   ][   ][   ][   ][   ][   ]
[   ][   ][ b&#8594;][  &#8594;][  &#8594;][  &#8594;][ &#8595; ]
[   ][   ][   ][ c ][   ][   ][ z ]   Does not go through c

2.
[ a ][   ][   ][   ][   ][   ][   ]
[   ][   ][ b&#8594;][  &#8594;][  &#8594;][ &#8595; ][   ]
[   ][   ][   ][ c ][   ][  &#8594;][ z ]   Does not go through c

3.
[ a ][   ][   ][   ][   ][   ][   ]
[   ][   ][ b&#8594;][  &#8594;][ &#8595; ][   ][   ]
[   ][   ][   ][ c ][  &#8594;][  &#8594;][ z ]   Does not go through c

4.
[ a ][   ][   ][   ][   ][   ][   ]
[   ][   ][ b&#8594;][ &#8595; ][   ][   ][   ]
[   ][   ][   ][ c&#8594;][  &#8594;][  &#8594;][ z ]   Goes through c

5.
[ a ][   ][   ][   ][   ][   ][   ]
[   ][   ][ b&#8595;][   ][   ][   ][   ]
[   ][   ][  &#8594;][ c&#8594;][  &#8594;][  &#8594;][z  ]   Goes through c


Therefore the probability of going from b through c from b 

is 2 ways out of 5, or a probability of 2/5

Edwin</pre>