Additional Odds for Flush Draw, by hitting two paris or three of a kind, as well - How to calculate Runner-Runner?

Question

Following an example from MIT Poker Course (slides can be found here, example starts at p9), where the odds are calculated from a drawing flush hand.

In the example, the situation is the following:

Hero (370$): A♥ T♥

Villain (370$): ? ?

Board (380$): 8♥ 3♥ K♣

Villian now goes all-in, and the course deducts, that it is correct to call, based on the 9 outs we have for completing a flush on turn or river. The lecture goes on, calculating, that any bet up to ~400$ would be profitable for us and should be called, based on the assumed 9 outs.

Neither on the slides, nor in the recorded lecture, any comments are given on what cards we put our opponent on. However, I assume villains hand is not that weak, as we need a flush to beat him. So for example, two pairs, or three of a kind (let's say three kings), he might have.

Nevertheless, in such cases, runner runner could also help us. For example, turn and river show both Aces. We would have a higher three of a kind. Or one Ace and one T show up. That would at least beat our opponent, if he has only two pairs.

So my question are:

• A) Can I assume higher odds, than just the ones based on hitting the flush? -> Which then would allow us to call an even heigher bet then the calculated maximum of 400$.
• B1) If so, how do I calculate or incorporate runner runners in a model? Like, how to treat runner runner cards mathematical?
• B2) If this is hard to calculate, is there a rule of thumb, easy to memorize way, for runner runners you can use quickly during a real game?

Answer 1

Usually runner-runner probability is so low that it won't affect the consideration of pot odds. Thus, in most cases, realistically, you calculate the direct odds. The only exception would be runner-runner flushes OR runner-runner straights (flop-->river), where the possibility is slightly less than 4% for each one, i.e. you can consider it as an extra out, as a rule of thumb.

Answer 2

Runner runner probability is simply multiplying the chance of getting one card in Turn by the chance of getting another card on the river.

Let's say you got a backdoor flush possibility. There's 10 flush cards left that you need on the turn, so that's 10/47; after getting the flush turn, you need the flush river, which is 9/46. 10/47 * 9/46 = 0.0416... So 4%. You can estimate with poker math (the outs times 2 rule) as well. 10 outs is 20%, 9 outs is 18%, 20% * 18% = 3.9%.