P&S is a tricky subject. Largely because people have very awkward intuitions about randomness and what it really means.
You’ll see ample evidence of it in any poker game. Here’s a typical Texas Hold ‘Em hand you see over and over and over in amateur games… one player’s holding three of a kind, while another’s got four cards to a flush, or maybe a draw to an open-ended straight, with one more card to come. Up to this point, they’ve got equal money in the pot, plus a few percent more from the other hands that’ve dropped out. The guy with the set pushes all in, and the guy on the draw calls. The final card hits him, giving him the hand, and everyone congratulates him for having played a "good hand".
The statistical reality is that the all-in bet offered him very little more than even money – each dollar he risked would get him just the same dollar from the other guy, plus a little fraction of that excess from the folded hands. So it’s only worth calling if he’s got a 50:50 chance of hitting that flush – otherwise it’s a stupid move.
But at that point, there are 46 cards whose whereabouts he doesn’t know (he knows the two in his hand and the four on the table). Only eight or nine of them will win the hand for him (eight if he’s drawing to a straight, nine if he’s drawing to a flush). So, based on the information available to him – we’re assuming he knows that he’s got to make his hand to win – it’s 4:1 or even 5:1 against making the hand.
Now, the analysis isn’t complicated. Any book on poker will explain how to count "outs" and figure "pot odds", and any experienced player will tell you that chasing straights and flushes in a no limit game is stupid, but if you play online, I guarantee you’ll see a really boneheaded move like that happen at least a couple of times in every game, followed by a round of praise from the table when it hits.
To make it worse, if someone points out that calling the all in was a really stupid play, the amateur players are all puzzled by it. How can you call it stupid? He won after all. But that’s just misunderstanding the nature of randomness.
Randomness – at least, above the quantum level – is all about information. To say something is "random" is to say that you don’t have the information to predict it. When you assign a probability to something, you’re actually making a claim as to how ignorant you are in that context.
The 4:1 odds against hitting a flush on the river are a probability – a 20% chance that the river card will be one of the nine cards you need. You arrive at that by admitting that the only thing you know about the locations of those nine cards are that the deck started with a certain collection, and you know the two in your hand and the four on the table. Any of the other 46 cards can be in any of the locations occupied by the cards that started out in the deck. The symmetries of the situation force you to the conclusion that each one is equally likely to be the next card. So if nine of 46 cards are winners, then you’ll find a winner 19.5% of the time.
But that’s not how our brains work, either. We’re wired to find patterns, even when there aren’t any. When you try to rely on intuition about probability, you’re almost certainly going to screw it up. The famous "Monty Hall" problem is a great example.
You’re on Let’s Make a Deal and Monty’s showed you three doors. You know there’s a big prize (say, a NEW CAR!) behind one door and joke prizes behind the other two (a lifetime supply of Turtle Wax, and a lawn-mowing goat). You choose one of the doors, and Monty, who knows which door hides the car, shows you what’s behind one of the other doors. Since he knows which one has the car, he can always show you one of the joke prizes. He then asks whether you want to stick with the door you initially chose or switch to third door. The question is, what is the optimal strategy? You can always stay with the initially chosen door, you can always switch, or anything in between.
Most people, relying on intuition, seem to come to the conclusion that it really doesn’t matter – that switching or staying still leaves you with a 50:50 chance at the car. A correct analysis shows that always switching is the optimal strategy, giving you 2:1 odds in favor of winning the car.
Intuition in probability and statistics will get you in trouble 95% of the time. :)
* * item2.1
# # item2.1
Welcome! OmniNerd's content is generated by nerds like you. Learn more.