Your options are:
Check and let him get a free card to either make his hand or improve his hand to a better losing hand.
Raise by more than a quarter pot and take down the pot here and now.
Bet a quarter and let him draw, or bet less than the quarter and let him draw.
I vote for betting a quarter each time. In each scenario he is less than 25% to make his hand, so you gain extra equity for each dollar you get him to commit to the pot above his justified odds
. Also, his implieds are zero. As it stands there are 28.5BB in the pot. You are ahead and are favourite to win from here.
If the pot is 28.5BB then betting a quarter is 7.125BB. He then calls 7.125BB for a final pot of 42.75BB
EV Situation 1 Qh 10h
(42.75BB x 0.8125) - (42.75 x 0.1875)
EV Situation 2 Ah 6c
(42.75BB x 0.9580) - (42.75BB x 0.042)
The payout could also change because if he catches a card that improves his own he cold pay us more, while we obviously wont pay him anything more if he makes his hand.
Another question, would you combine the two scenarios, giving him 11 outs, in order to gauge EV or would you do each one individually? If you combine them then the quarter bet is still correct and the EV is
(42.75 x 0.7756) - (42.75 x 0.2244)