Pokemon Cards and Probability

[mathgrant]

I had a dream that I was playing the Pokemon Card Game with a statistician. He left the room while we were playing, and I was losing. I was tempted to look at his cards, but decided to do something with my own cards instead. I had Weezing, Squirtle, and Pikachu on my Bench. I shuffled them thoroughly (tough to do when you only have three cards!) and chose one at random. Then I got out a sheet of paper with the numbers 1 through 6 and wrote next to the 1 the name of the Pokemon I chose. I did this five more times, writing each new Pokemon next to the next number.

Now, as we all know here at the GL, that would be a great technique for generating a random number between 1 and 729, but that wasn't what I was interested in. I then rolled an evenly weighted 6 sided die and whatever Pokemon was next to that number, that would be the Pokemon I would use after my current one was Knocked Out. But before I could see the number, I woke up!

Oh no! I thought. Which one was I about to choose? Well, I couldn't be totally sure, I never saw the number, best I could do was remember what Pokemon appeared in each slot on the paper and use simple statistics to estimate what could've happened. But oh no! I couldn't remember anything! Except that each Pokemon was in at least one of the six slots. And Weezing appeared at least twice. But that was it.

I then proceeded to calculate, with only that information, the probability that I would've chosen Pikachu. I went back to sleep and had the dream again. The statistician seemed to be a better psychic than Miss Cleo; not only did he know what I did when he was away, he even knew what had happened when I woke up. I told him the probability I had calculated, and he was impressed at my statistics skills. He then resigned the Pokemon game and started having a talk with me about statistics instead.

That was the best dream I'd ever had in my life. Now, two questions:

1. What was the probability of me choosing Pikachu?

2. Hidden somewhere in this problem there is a reference to a famous graph theory chestnut. What is it?

[This message has been edited by mathgrant (edited 02-13-2002 03:56 PM).]

[ZutAlors!]

1. How about 15/54?

[Chuck]

1. Maybe 7/24?

[mathgrant]

Number 1 was answered correctly by: Chuck

[Chuck]

Graph Theory has chestnuts? That's hard to believe.

[mathgrant]

But it does. Can you find one?

[ZutAlors!]

Ok, what am I missing?

Step 1: "...each Pokemon was in at least one of the six slots. And Weezing appeared at least twice" = without loss of generality, assume slots 1-4 are WWSP.

Step 2: Slots 5 and 6 are equally likely to be any of the Pokemon. Therefore, there are nine possible combinations, all equally likely: WWSPWW, WWSPWS, WWSPWP, WWSPSW, WWSPSS, WWSPSP, WWSPPW, WWSPPS, and WWSPPP.

Step 3: Given any of the above combinations, the probablity of choosing Pikachu is simply (#P/6), which is, in order, 1/6, 1/6, 2/6, 1/6, 1/6, 2/6, 2/6, 2/6, and 3/6.

Step 4: Total probability is then (1/9)*(1/6 + 1/6 + 2/6 + 1/6 + 1/6 + 2/6 + 2/6 + 2/6 + 3/6) = 15/54 = 5/18

[ralphmerridew]

I figured out what the chestnut is, and I will say that it involves the three pokemon cards. Figuring out the chestnut will require some knowledge of pokemon.

[extropalopakettle]

ZutAlors!, I haven't done the math, but I'm suspicious of step 2.

Let's say I perform a large number of trilas where I randomly fill the 6 slots with W's, S's and P's. I only consider those trials that have at least 2 W's and at least 1 each of S and P - all other trials are discarded. Now remove 2 W's and 1 S and 1 P. There are nine possible combinations for the remaining two cards - WW, WS, WP, SW, SS, SP, PW, PS, PP. But I'm not sure all are equally likely. The reasoning being along the same lines as, if I flip two coins, and at least one is heads, the probability of two heads is 1/3 (not 1/2).

[tigg]

I got the exact same answer you did, Zut.
Did we make a bad assumption?

[tigg]

I'm starting to think your step 1 was our bad assumption. Looks like there IS a loss of generality.
extro's example of the two coins makes a good point.

[ZutAlors!]

Ah, OK. This is akin to the famous "A couple have two children; at least one daughter; what are the chances they also have a son?" I should have realized it wasn't as easy as I thought.

[mathgrant]

So, what's the graph theory chestnut I'm looking for?

[mole]

The one with the three houses and the gas and water and stuff? That's the only one that comes to mind - can't see it here yet though

[mith]

weezing=gas, squirtle=water, pikachu=electricity

[mole]

Gas isn't a type of pokémon, is it? Maybe all these houses want to be connected to the poison supply...