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Big Fish · April 14th, 2009, 10:06 AM

Being able to determine expectation value (EV) situations quickly at the table is extremely important so that one can perform intelligent analysis during play. Since EV calculations can become quite cumbersome, one way to do this is to shift our thinking from probabilities to odds. Odds are nothing more than a different way of expressing probabilities. Where probabilities refer to the number of outcomes of interest with respect to the total number of possible outcomes, odds refer to the number of outcomes of interest with respect to the number of outcomes not of interest. For example, the odds of drawing an Ace from a full deck is 4/52 or 1/13 (one Ace for every 13 draws). Expressed in terms of odds, the odds in favor of drawing an Ace is 1:12. Therefore, 13-1=12 outcomes are not Aces making the odds 1:12. You can also talk about the odds against drawing an Ace, 12:1, the reciprocal of the odds in favor of drawing an Ace. When evaluating decisions in poker, one is usually considering the odds against.

With this understanding of odds, calculating EV becomes much easier. Say I give you $10 for every time you draw an Ace and you give me $1 for every time that you draw any card other than an Ace. On average, if we were to play 13 times, you would have to pay me $1 twelve times and I would pay you $10 once. In other words, you would pay me $12:$10, meaning that I make $2 per 13 draws.

To look into this a little closer, the payout odds for this situation are $10:$1. Payout odds tell you the amount of money you stand to win given the amount of money you wager. Meanwhile, the odds pertaining to the random event itself (12:1 in this case) are called the event odds. So when does this game become fair? I would have to pay you $12:$1 to make the game fair and if I paid you more than $12:$1, you’d be profiting. The key concept here is that you come out ahead if the payout odds are larger than the odds against the event you’re wagering on.

Lets say that you are playing at a casino that offers a side bet paying 7:1 on a bet that the flop will contain 3 red cards. Should you make this bet? Of course you are forced to make this bet before you are dealt cards. So the deck contains 52 cards, thus there are (52*51*50)/(3*2*1)= 22,100 possible flops. Since half the deck (26 cards) are red, out of those 22,100 flops, there are (26*25*24)/(3*2*1) = 2600 flops that contain three red cards. Therefore the odds against you flopping three red cards are (not all red)/(all red) = (22100-2600)/2600= 15/2 or 7.5:1. Thus for every $7 you make, you are giving back $7.50. You are losing .50 for every $8.50 wagered (the odds in this case are considered in cycles of 8.5 trials since 7.5+1=8.5).

So how does this apply to poker you might be asking? Well, it’s the turn and you have to hit a flush draw to win. Your opponent bets $100 into a pot of $300. Do you have the odds to call here? To figure this out you need to know the odds of hitting your flush with one card to come. There are 4 cards on the board and 2 in your hand, meaning that you know 6 cards in the deck: 52-6=46 cards left in the deck. Out of those 46 cards, 9 of them complete your flush. Therefore the odds against you completing your flush are (46-9):9=37:9 (reduced down to 4.11:1). Meanwhile the pot has $300+$100=$400 and you have to call $100. Therefore the pot odds are 4:1. Since the odds of you hitting the flush are worse than 4:1 you should fold. But what if you decide to make the call anyway. Since the odds of hitting a flush are 37:9 in 46 trials the payout odds are 37($100):9($400)= $3700:$3600. So in 46 trials you lose $100 if you make this call or $100/46=$2.17. Another way to approximate this would the fact that you need an approximate $411 return for every $100 you invest. Therefore, if you call you lose about $11/5.11=$2.15 per call because you put up this imaginary $11 over the span of 5.11 hands. The 5.11 comes form the fact that our approximate odds of hitting the flush are 4.11:1, meaning that there are 4.11+1=5.11 total trials to consider.

Ed Hill · April 14th, 2009, 05:48 PM

Quote:

Originally Posted by Big Fish

So how does this apply to poker you might be asking? Well, it’s the turn and you have to hit a flush draw to win. Your opponent bets $100 into a pot of $300. Do you have the odds to call here? To figure this out you need to know the odds of hitting your flush with one card to come. There are 4 cards on the board and 2 in your hand, meaning that you know 6 cards in the deck: 52-6=46 cards left in the deck. Out of those 46 cards, 9 of them complete your flush. Therefore the odds against you completing your flush are (46-9):9=37:9 (reduced down to 4.11:1). Meanwhile the pot has $300+$100=$400 and you have to call $100. Therefore the pot odds are 4:1. Since the odds of you hitting the flush are worse than 4:1 you should fold. But what if you decide to make the call anyway. Since the odds of hitting a flush are 37:9 in 46 trials the payout odds are 37($100):9($400)= $3700:$3600. So in 46 trials you lose $100 if you make this call or $100/46=$2.17. Another way to approximate this would the fact that you need an approximate $411 return for every $100 you invest. Therefore, if you call you lose about $11/5.11=$2.15 per call because you put up this imaginary $11 over the span of 5.11 hands. The 5.11 comes form the fact that our approximate odds of hitting the flush are 4.11:1, meaning that there are 4.11+1=5.11 total trials to consider.

What you are quoting here is called express odds, that is; there is no more betting involved. With a flush draw, you also have to take into account implied odds. With the flush draw example you quoted, it costs you $100 to draw and if you miss that is all that it is going to cost you, but if you hit, are you going to win any extra money on the river?

Big Fish · April 15th, 2009, 09:30 AM

My intent for this post was to just show some basic math behind calculating odds. I wasn't trying to get into the meat and potato's of it all with this post. But yes, if you did make the $100 call and catch your flush on the river and was then able to get a call of $150 out of your opponent on the river then your pay out odds change from $400:$100 to $550:$100 making the $100 call on the turn now a +EV call.

new2Poker · September 13th, 2009, 01:12 PM

I dont have much expirence on this matter how ever im sure if u tried with the right methods u would suceeds

mpourkos · September 14th, 2009, 03:11 PM

All these maths are pretty easy on online poker! There are a lot of tools and online calculators

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