Hero: 3♠3♥
Villain: A♦K♦
Board: 5♦3♦T♠
There are 45 unknown cards. On the turn 7 of them give hero 100% equity and villain 0% equity. The rest give varying amounts of equity to hero and villain. We can look at those in terms of how many river cards will win. We group cards into categories that have similar characteristics (flush, no draw, gutshot draw, etc.). It looks like the below.
- Turn Full House: 7/45 -- Hero wins 44/44 rivers.
- Turn Flush: 8/45 -- Hero wins 10/44 rivers (quads and boats)
- Turn No Flush Q,J,2,4: 12/45 -- Hero wins 34/44 rivers (non-straight and non-flush cards).
- Turn No Flush A,K: 6/45 -- Hero wins 36/44 rivers (non-flush cards)
- Turn Other No Flush: 12/45 -- Hero wins 37/44 rivers (non-flush cards)
Then we add the probabilities for each hand we win. 7/45 * 44/44 + 8/45*10/44 ... etc. And that's the total probability to win. Since the denominator is the same, we just multiply the numerators and add all the products. Then we can divide by 45*44.
(7*44 + 8*10 + 12*34 + 6*36 + 12*37) / (45*44) or 1456 / 1980 ~= 73.5% equity for hero and ~= 26.5% for villain.
* These cards offer gutshot straight draws on the river.
* Removes an otherwise safe river card from play on the turn.