So I like puzzles and games and I really love analyzing them. Its the science of side of me, what can I say? Im also trying to become a better writer and the universal tip from all writers on becoming a better writer is to write daily. So Ive been writing 1 new idea a day. Well, the other day I became enraged by a Euchre game app on my phone that drove me insane and I got a little side tracked. I figured my family might enjoy/get this. Games are a sacred part of human nature. We love games. We like the challenge, we like the puzzle, we like the competition. So when you cheat at a game, and I mean any game, we will metaphorically kill you (in some places they actually kill you if you gave them the chance). This sanctity applies in different degrees to different games depending on the person you’re talking too. In general, the longer the tradition for you and your family before you, the more sacred. For me and my family, one of those games is Euchre. Euchre is a simple card game played with a deck of 24 (Ace through 9) with 4 players in teams of 2. The objective is to get to a certain number of points before the other team. You get points by taking “tricks,” If you take all 5 of the tricks where each player throws in a card, you get 2 points. If you took 3 or 4 of the 5, you get 1 point. If you took less than that, you got “euchred” and the other team will start yelling it in your face. It’s a real bonding moment. The other way you can score points is by “going alone.” This is where your partner won’t play his cards to help you. The scoring is the same unless you take all 5 tricks and then you get 4 points instead of 2. You can play aggressively or conservatively depending on the situation, just don’t cheat. If the first person throws in a diamond on any given trick and you have a diamond, you have to follow suit and play that diamond. If you don’t that’s called reneging, which is followed by more yelling in your face and more family bonding time. There are other ways to cheat, but you get the point. Now since Euchre is pretty straight forward, it lends itself to being a pretty good phone app. It also moves faster because crazy grandma isnt playing and she isnt conveniently “forgetting” the rules all time. This means you might notice patterns and see certain situations more frequently than you might otherwise see them. Take for example this situation, which Ive played 5 times now, where you are to the left of the dealer and you have 4 of the highest trump cards and a king of another suit. Every time I see this hand or something very similar, I’m going to call it up and go alone to try and get 4 points. So then what are the chances of me taking all 5 tricks and earning my 4 points? Surprisingly small it seems according to the game on my app. All 5 times Ive tried this, I have failed. This seems odd to the scientist in me and so I decided to run the numbers. Assuming you are to the left of the dealer, with the 4 highest trump cards (lets say you called up spades) and you play those 4 cards first, you are guaranteed to take the first 4 tricks. It’s also guaranteed that youve pulled out all the other trump cards. Now lets say that the last card in your hand is the king of diamonds. The only card that can beat you at this point is the ace of diamonds. So if someone on the other team has the ace of diamonds in their hand at the last trick you lose. But what’s the probability of that happening? Lets go back to hand as it was being dealt. All players receive 5 cards with 4 left in the stack. I know that ace of diamonds is obviously not in my hand so we can discount my 5 cards. Now there are 10 possible places in my opponent’s hands that the ace of diamonds could be and 5 in my partners hand, which won’t be played. I know it’s not the card on top of the stack because that was a spade, but it could be one of the other 3 cards in the stack. If you’re keeping track, that means that my opponent has a 10/18 or 5/9 or 55.5% chance of having the ace and beating me with it. But not so fast. There’s also a very small chance that the dealer (who in this situation is my opponent), had the ace of diamonds in his hand when he picked up the spade on top of the pile and discarded the ace of diamonds to the stack as he replaced it with the spade in his hand. The chance of this happening is very small. The only reasonable hand in which this could happen is if the dealer had the remaining 2 trump already in his hand and three aces. In this case he would have to discard an ace to pick up the trump and so he would have a ⅓ chance of discarding the card that could beat me, but what is the chance of him having that hand in the first place. There is only 1 hand of 5 very specific cards that faces this ⅓ chance. The chance of having any one of those 5 cards in the dealers hand is individually 5/18 discounting my cards, and the card on the top of the stack again. So we multiply 5/18 or .277 by itself 5 times to give us a rough idea of the odds of this particular hand happening in the dealers hand which is again a ⅓ probability . The odds of this hand occurring randomly are .0016 or .16% (It’s actually probably closer to .00068587 if you account for variations in cards already in your hand and cards remaining to be dealt, but we’ll let it slide). The odds of this happening in the dealers hand specifically is ⅓ that or .00054 or .054%. The odds of the dealer discarding that specific Ace are again ⅓ that or .00018 or .018%. Needless to say, the chances of the card that beats my king being discarded back on top of the stack from the dealer’s hand are very small. .00018 to be specific. So let’s assume that didnt happen. Is there anything else which might save me if the ace was in the other team’s hands? Well if for some reason they discarded it before the last trick was played, it couldnt beat my king as it wouldnt be in their hand when they needed it. So what would get a player to throw an ace away, which is the highest card outside of trump? Well if the player had 2 aces or even 3 aces, they would be forced to decide which they would keep the longest in hopes of beating a lesser card of the same suit. So assuming he already has the ace of diamonds, there is a 4/18 or .222 chance of him having the ace of clubs and a .222 chance of him having the ace of diamonds. There is then a .049 chance of him having both or 4.9%. If he has both, then there is a ⅔ chance of him discarding the ace of diamonds early or a total chance of .033 or 3.3%. If he has just one of those aces he has ½ chance of discarding the ace of diamonds or a total chance of .111 or 11.1%. Adding those 2 probabilities together gives you an overall probability of .144 or 14.4% chance that if they did have the ace of diamonds, they would discard it prematurely. This leaves out one factor. Good Euchre players count cards. There is a chance that my opponent will count the cards in his hand and in his partners hand and notice that perhaps 4 or 5 of the clubs were played in the first 4 hands and only 2 of the diamonds. This increases the probability that he will keep the ace of diamonds instead of the ace of clubs or hearts depending on the situation. The chance of this happening are good, but it’s hard to determine the effectiveness of the strategy through the myriad different situations. For argument’s sake let’s say that this improves my opponents odds of picking the right ace by 25%. This means that it reduces the odds of premature discarding of the ace of diamonds from .144 to .108 or 10.8%. If we add this odds to the odds of my opponent throwing it back on the stack it gives us a .10818 or 10.818% chance that he will have the ace but then lose it. If we assume that from the get go they had a .555 or 55.5% chance of having the ace of diamonds in the first place and we multiply that by the inverse odds of losing it we get a value of .495 or 49.5% chance that my opponent will beat me and prevent me from getting my 4 points. So given those odds, what is the chance that he will beat me in this or a nearly identical situation 5 times? You simply multiply the odds by itself 5 times and you get a number like this: .0297 or roughly 3%. That means that it is statistically unlikely for this to happen. In other words, my phone app cheats like my crazy grandma (and a few other relatives).
Posted on: Mon, 04 Nov 2013 20:17:58 +0000
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