We begin with:

First box         Second box
$5+$1+$1+1$         $5+$1

Junior gets to take out a bill from the first box without looking and put it in the second box.

Then we have either of these cases:

Case 1, with a probability of 1/4

First box         Second box
$1+$1+1$           $5+$5+$1

and

Case 2, with a probability of 3/4

First box         Second box
$5+$1+$1           $5+$1+$1

he then gets to take one bill from the second box

So,

P($5) = P[(case 1 AND drawing a 5$) OR (case 2 AND drawing a $5)]

= P(case 1)×P(drawing a $5 given case 1) + P(case 2)×P(drawing a $5 given case 2) =

(1/4)(2/3) + (3/4)(1/3) = 5/12

Edwin