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Should You Wait?

For instance you are playing in a ranking hold'em or lowball tournament and find yourself down to a very short stack. Assume you have enough to call the big blind. And there are only one or two hands before you will be forced to blind it. Many of them ask if it is correct to loosen up your calling requirements. They suggested that as you have to blind it anyway on the next hand, you should play any hand that is better than the average random hand you will get on your blind. While some others say that the upcoming hand makes no difference. It should not influence you to take the worst of it on the current hand.

While thinking over to this problem, it is realized that stated mathematical question can be solved to give some insight into this problem. For instance you have $100 in your pocket with no way of getting more for the time being. In another five minutes, you will be forced to bet this $100 with someone on a sports event or something other that. You can only get out of his bet if you lose the $100 before he shows up. (However, if you have increased your bankroll, your bet will still be exactly $100.)

The question comes what should you do if someone offers to bet $100 on something right now? If you take it, you will either wind up with $200 ($100 of which you will have to bet back in five minutes so that your final bankroll will be either $100 or $300) or you will go broke, which eliminates your compulsion to make the second bet.

Before determining whether to take the first bet or whether to pass it and just wait to make the second bet, we have to know the chances of winning each bet. Call your chances of winning the first bet "X" and your chances of winning the second bet (if possible you get the chance to make it) "Y."

For finding some solution to this problem, we will calculate the expected value of your bankroll if you take the first bet and also if you pass it.

The second alternative is easy to figure. If you pass the first bet, and only make the second bet, the expected value of your playing bankroll is just $200 Y. What about if you do take the first bet? Now you will wind up with either $0 (chances of this are [1-X], $300 (chances of this are XY) or $100 (chances of this are X[1-Y]). So the expected value of your bankroll if you take the first bet is:

0(1-X) + 300XY + 100X [1-Y]

This reduces to:

200XY + 100X

the expected value of your bankroll if you accept the first $100 bet and then make the second bet if you win it. Notice that both alternatives give you the exact same expected value if

200Y = 200 XY + 100 X

Solving for X:

200Y = X (200Y+100)

X = 200Y / 200Y+ 100

= 2Y / 2Y + 1

So if the possibility of winning the first bet is greater than:

2Y / 2Y + 1

where Y is the possibility of winning the second bet, then you should accept it.

For example, if you chances of poker winning the upcoming bet is 40 percent then you should gamble on preceding bet if your chances are (.8) / (1.8) which is about 45 percent. If your upcoming bet is ten percent the preceding bet should have a possibility of (.2) / (1.2) or about 17 percent to make it worthwhile.

These figures show that if your bet does perhaps have little chance, it is right to make an earlier bet with less the worst it. On the other hand, this earlier bet should be more than just better than the upcoming one. (Notice that if the upcoming bet is exactly 50 percent, then the possibility of winning the first bet must be greater than:

2 (.5) / 2 (.5) +1

which is one half. This agrees to some common sense.)

What may not agree to this common sense is what you do if your upcoming bet has a greater than 50 percent chance. Now taking this first bet will risk your opportunity to make this second good bet. Does this mean you should pass up the first bet even if it has little bit the best of it? By using the formula, we see, for instance, that if the second bet has a 60 percent chance of winning the chances for the first bet must be greater than (1.2) / (2.2) or about 55 percent chance. If the second bet has an 80 percent chance, the first bet should be (1.6) / (2.6) or about 62 percent. Therefore it is correct that it might be right to pass up a bet with slight best of it if the upcoming bet has a lot the best of it. However, you should never pass up a bet as more than 2-to-1 favorite (if the second bet is 100 percent, the first bet by this formula must just be above (200%) / (300%)).

These outcomes have great importance, not just to tournament play either. If you are one of those people who consistently risk going broke with slight the best of it even when there are better bets over the horizon, you may want to think again over to this chapter.

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