Pedigree Probability of Autosomal Recessive Trait
Starting with the left hand side of the diagram:


III:2 is definitely a carrier (Tt) as one parent (II:2) is affected (tt).
III:1 is also definitely a carrier (Tt) as when mating with III:2 they produce an affected (tt) offspring (IV:1)
This means that we can work out the possibilities for IV:4 as we know the parent genotypes.  It follows the standard arrangement for two carrier parents giving the options of:

TT (1/4)
Tt (2/4 = 1/2)
tt (Normally 1/4 but in this case 0 as individual not marked as affected). 

Therefore for this scenario, the probabilities for IV:4 are :

TT (1/3)
Tt (2/3)



Now if we look at the right hand side of the diagram.


IV:5 is definitely a carrier (Tt) as one of their parents (III:5) is affected.  


This gives two possible Punnett squares to be examined:

|-------------------------------------------------------------------------------|
|                                        ♂ (IV:5)                               |
|                          T                                   t                |
|            -------------------------------------------------------------------|
|           |                                |                                  |
|         T |              TT                |                Tt                |
|           |                                |                                  |
| (IV:4)    |-------------------------------------------------------------------|
|    ♀      |                                |                                  |
|         T |              TT                |                Tt                |
|           |                                |                                  |
|-----------|-------------------------------------------------------------------|


This gives nil affected offspring so we can disregard this option for your question (as we are ONLY looking for scenarios which produce affected individuals).

Therefore the alternative is:

|-------------------------------------------------------------------------------|
|                                        ♂ (IV:5)                               |
|                          T                                   t                |
|            -------------------------------------------------------------------|
|           |                                |                                  |
|         T |              TT                |                Tt                |
|           |                                |                                  |
| (IV:4)    |-------------------------------------------------------------------|
|    ♀      |                                |                                  |
|         t |              TT                |                tt                |
|           |                                |                                  |
|-----------|-------------------------------------------------------------------|


Giving 1/4 affected offspring.  

As mentioned above, in order to have affected offspring then IV:4 must be Tt.  There is a 2/3 chance of this being the case.  If this is the case, then there is a 1/4 chance of the child being tt. 

Both conditions need to be true for this to happen so we multiply the fractions:

2/3 * 1/4 = 1/6