#implied odds

In this series we established a model for directly considering important factors for determining the implied odds needed for calling a current street bet where the pot odds are insufficient. If P$ is the pot after villain bets on the current street, C$ is the amount hero has to call and F$ is the amount hero will bet on the next street if he hits his outs, the implied odds equation is as follows:

IO = (P$ + F$)/C$

To determine F$, the future bet requirement to achieve +EV, we have the following:

EV =(1-H)(-C$)+ H(C(W(P$+F$) - (1-W)(C$+F$))) + H(1-C)P$

F$ = (C$ - HK$(1+CW- C)) / (CH(2W- 1))

where

K$ = P$ + C$, the pot after all bets are made on the current street

H = the probability hero will hit his outs on the next street

W = the probability hero will win the hand if he hits, bets and villain calls

C = villain call probability when hero makes a future bet

For example, for a set mining situation with a low to middle pair, the chance for a hit is about 12%, meaning if you hit and win, you need pot odds of 88 to 12 or 7.3 to 1. Given that you don’t have these odds, you consider the possibility of hitting the set on the flop and winning big. Recognizing that you may not win with a hit, villain may not call if you do hit or the effective stack you might win is not very big, you often see a comment that you should have implied odds or 2 to 3 times the minimum odds, which would be about 15 to 20 to 1 for set mining. The model we propose is an attempt to quantitatively include these factors so that there is mathematical justification for the required implied odds.

Villain Calling Percent Variations. For simplicity both in concept and exposition, the results we provided in the previous posts had a fairly strong assumption, namely that villain always calls hero’s bet. We now will attempt to show some results where we no longer make this assumption. The following table shows the minimum implied odds for various combinations of Call%, (C), C$, Hit and Win. In every case the implied odds requirement goes down as Hit or Win increase or C$ decreases. This is to be expected for hit or win increasing means hero’s equity is increasing so the future bet requirement decreases. On the other hand, as C$ gets bigger, hero’s risk is greater, thus the reward as represented by the implied odds has to be greater.

For the Call%, the implied odds also decrease for all the table values we considered, Note that the W values, the chance of winning if hero hits his outs, are greater than 50% for all cases. A player with a greater than 50% winning chance normally would prefer a villain call rather than a fold and that is reflected in the table values.  However, we did not include a large range of C$, W and H combinations so this is not always true as there can be instances where greater villain calling does result in a larger future bet and implied odds requirement, usually for low hit values and/or large C$ values.

Example. Assume hero has to call a bet that is 50% of the pot after villain bets on the current street. If the expected hit probability is 20%, such as would be the case for a flush draw, and the win probability is 75%, the following implied odds are required for the following call probabilities:  Call= 10%, IO = 43.5. Call = 50%, IO = 11.5. Call = 100%, IO = 7.5. Because of the high equity given a hit (W=75%), we see that the implied odds and associated future bet requirement decreases as the call probability increases.

Model Application and Summary