More Math Shortcuts (Inducing a bluff)

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This idea comes from trying to answer a question in Exitonly's post

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I love it how an interesting post always seems to come up during the same day's play. I just had this hand a short while ago. It seems to mirror the concept you are getting at.

Hand #1045xxxx-16 at XXXXpmA-010 (No Limit tournament Hold'em)
Started at 22/Dec/05 18:16:42

PlayerA is at seat 0 with 1805.
PlayerB is at seat 1 with 1660.
PlayerH is at seat 2 with 1410.
PlayerC is at seat 3 with 3025.
Villain is at seat 4 with 1540.
PlayerD is at seat 5 with 1770.
PlayerE is at seat 6 with 1055.
Hero is at seat 7 with 1355.
PlayerF is at seat 8 with 1470.
PlayerG is at seat 9 with 1410.
The button is at seat 8.

PlayerG posts the small blind of 10.
PlayerA posts the big blind of 20.

PlayerA: -- --
PlayerB: -- --
vernally: -- --
PlayerC: -- --
Villain: -- --
PlayerD: -- --
PlayerE: -- --
Hero: Qh Ah
PlayerF: -- --
PlayerG: -- --

Pre-flop:
PlayerB folds. PlayerH folds. PlayerC folds.
Villain raises to 70. PlayerD folds.
PlayerE folds. Hero calls. PlayerF folds.
PlayerG folds. PlayerA folds.

Flop (board: Jc Ad 5h):
Villain bets 170. Hero calls.

Turn (board: Jc Ad 5h 3d):
Villain checks. Hero checks.

River (board: Jc Ad 5h 3d Ac):
Villain bets 225. Hero calls.

Showdown:
Villain shows 9c 9s.
Villain has 9c 9s Jc Ad Ac: two pair, aces and nines.
Hero shows Qh Ah.
Hero has Qh Ah Jc Ad Ac: three aces.

Hand #1045XXXX-16 Summary:
No rake is taken for this hand.
Hero wins 960 with three aces.