Psyprobs 1

The subtle and celebrated art of psychological warfare has been part of almost every war, battle, skirmish and brawl in history. These days they make bloodhounds bark into the faces of Guantanamo Bay prisoners and market it to kids as ‘PSYOPS.’ I am not going to burn effigies of bloodhounds like the politically aware would, but I will talk about the concept of Psychological Probabilities that I have been forced to explore since I started playing online poker. I have more questions than answers and the reader(s) will probably dismiss Psyprobs as just conditional probabilities in disguise. Consider the following scenario:
You’re engaged in a sophisticated game of strategy and chance where the bloke tosses a fair coin and you win if its heads and cough up the cash if its tails. You shift through your mental gears and conclude ‘50-50′ until the bloke throws a psychological spanner. “I should tell you that the last 100 tosses have all landed tails.” Wocha gonna do? Are you gonna remind yourself that a coin toss is an independent event and you have an even chance on the next toss regardless of prior (history?) Will you conclude that since a ‘balance’ has to maintained in the long run, a heads is just around the corner and double your bet? Or will you just figure it ain’t your day and pack it in? The fact is you’ve been given some extra information. How are you going to use it? Should it bother you that the coin landed 100 consecutive tails? Can (and should) this run of tails be factored into your decision making in a precise mathematical way? Sometimes, what seems to be a Psyprob scenario is just conditional probability in disguise. The classic Monty Hall Problem is an example. I will not go through the setup and solution. This Wikipedia entry does a good job.
I believe that the first example is just a standard coin-toss and the ‘extra’ information should not come in the way of your gambling fortune. The Monty Hall Problem is clear-cut conditional probability in action. Here’s yet another scenario which may require something more than conditional probability. Then again, maybe not. You decide.
So you’re playing poker heads-up. You’re dealt a pair of Kings. The flop (the first 3 of the 5 community cards) is a pair of 7s and a Queen. The suits are such that a flush isn’t a factor. You make a serious bet, your opponent goes all in. You figure maybe he has a pair of As and are ready to fold when he shows you one of his cards: a Queen. Agreed that there are two more cards to follow but, as it stands, can you work out the odds of your having the winning hand?
Well…he has the better hand if his other card is a 7 or a Q. Should this be the sole consideration when you calculate odds or should you think in terms of, “is he showing me the weaker/stronger card?” The former is conditional probability while the latter has Pysprobial undertones. Are both the approaches one and the same? It is tempting to say Yes! If he’s showing the weaker card you calculate the odds and add them to the odds if he’s showing the stronger card and it should tally with the conditional probability calculation. I suspect, however, that each component in this addition should be ‘weighted’ in some way to reflect the psychology of your opponent in showing the weaker/stronger card. Think about it. I’ll post some actual calculations later. All I know is that Stu Ungar wouldn’t have put up with this kinda rubbish.