Calling range and optimal bet sizes (maths peoples help)

No-Limit Hold'em, $0.25 BB (6 handed)

UTG ($26.85)
MP ($52.84)
CO ($36.09)
Button ($24.27)
SB ($41.12)
Hero (BB) ($25.85)

Preflop: Hero is BB with A, 9
3 folds, Button bets $0.75, 1 fold, Hero calls $0.50

Flop: ($1.60) 6, 3, 7 (2 players)
Hero checks, Button checks

Turn: ($1.60) 7 (2 players)
Hero bets $1.50, Button calls $1.50

River: ($4.60) A (2 players)
Hero bets $3.40, 1 fold

Total pot: $4.60

Results:
Hero had A, 9 (two pair, Aces and sevens).
Outcome: Hero won $7.70

Hand isnt mine but im trying to help a friend and also get my head around this calculation

Dont worry so much about the given ranges vs bet size its the math or way we figure it out more I am interested in!

Calling Range for turn

AK
A6,K6,Q6,J6,96,86,56,46
9T,98
97,78,57
88,55,44,22

Bet Size - $2.40 Calling river range

AK, A6, 97,78,57
88,55,44,22

Bet Size - $3.40 Calling Range

A6, 97,78,57

How do we work out missed value from small pairs when we bet $2.40 but value cut ourselves when betting $3.40 on river and therefore given total calling range which is the most +EV bet size.

I dont care about the ranges that much if you think they are too tight or loose or w.e. just want to know how to work these calculations for future references

(looking at stu, fp, chuck, other math guys)

EV = F(P) + (1-F)[W(P+B) - (1-W)B]

F = %opponent folds
P = pot size before bet
B = bet size
W = %you win (Pokerstove equity vs his range)

You also need to know your opponents total range before you make the bet so as you can get his fold % (bigger the range smaller the fold %).

W changes as F changes as obviously the stronger his range the lower your equity.

Thanks Jurn. This was a hand I played and Jurn commented that my bet was too large (most likely correct looking at it). For the sake of completeness, only had 27 hands on villain (so almost readless) however, his stats were: 52/41. So at least we can say he wasn't being shy.

cheers bro!

the % he folds is the combos of folds out of total combos yeah?

Also is the W of his total turn call range I guess

....