33 flops set board turns ugly

[Everlong]

BB is a product of watching poker on TV. Probably mostly WPT final tables and thus loves any ace and will gladly take it un-improved to showdown. He is a ship lost at sea post-flop. This is my very sophisticated read after about 35 hands [img]/images/graemlins/smile.gif[/img] ....

My concern is, even though I guessed right here, is checking behind with position here a leak? After the hand I felt I should have bet the river..

Hold'em (6 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: 2+2 Forums)

Preflop: Hero is MP with 3[img]/images/graemlins/diamond.gif[/img], 3[img]/images/graemlins/spade.gif[/img].
<font color="#666666">1 fold</font>, <font color="#CC3333">Hero raises</font>, <font color="#666666">3 folds</font>, BB calls.

Flop: (4.40 SB) 2[img]/images/graemlins/club.gif[/img], 3[img]/images/graemlins/heart.gif[/img], J[img]/images/graemlins/spade.gif[/img] <font color="#0000FF">(2 players)</font>
BB checks, <font color="#CC3333">Hero bets</font>, BB calls.

Turn: (3.20 BB) 5[img]/images/graemlins/club.gif[/img] <font color="#0000FF">(2 players)</font>
BB checks, <font color="#CC3333">Hero bets</font>, BB calls.

River: (5.20 BB) 4[img]/images/graemlins/diamond.gif[/img] <font color="#0000FF">(2 players)</font>
BB checks, Hero checks.

Final Pot: 5.20 BB