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

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?
• If villain has a set, runner runner Ace would give them a full house, so that possibility can't be counted as full outs.
– Herb
Jul 27 '18 at 15:21

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.

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%.