Ex Sitibus: A Super Mario Triptych « The Journal of Cartoon Overanalyzations

Ex Sitibus: A Super Mario Triptych

mario_bar

Just to temper our postmodern ennui, we’re shaking things up a jot here at J. Cart. Overanal. Below we have three over-analyzationy links concerning Mario, who despite being a workaday fungal-plumber is really quite the Homo Universalis.

One: a brief, unwieldily image-based thesis on the symbology of Super Mario Bros., including a blunt, presumptive explanation for the meaning of life. Just click on the image to expand it legibly. (The true origins of this thesis have defied our gumshoeing; if anyone knows we’ll link to it properly.)
Two: another masterpiece by the quasi-intellectuals at McSweeney’s, in the approximate vein of these other ones. Here we have an editorial by Dr. Mario himself, lambasting the Mushroom Kingdom’s corrupt and inefficient health care system.
Dr. Mario Weighs in on Universal Health Care > Catena Ex Situ
Three: yonder at The Minus World, we have a cleverly-conceived, confusingly both underwritten and overwritten, and woefully trying-too-hard-to-be-funny report about the frequency of Princess actualization.
Study Shows The Princess is in Another Castle 7/8ths of the Time > Catena Ex Situ

Addendum: The calculation to find the Princess Probability isn’t even accurate. Taking into account the Warp Zones, there are many different paths Mario might take during the course of the game, each with a different total probability of the Princess being in a castle, hereafter referred to as $P(Princess)$ .

In World 1-2, there is a Warp Zone to Worlds 2-1, 3-1, and 4-1. In World 4-2, there are two Warp Zones, to 5-1, 6-1, 7-1, and 8-1 in toto. So let’s break the Mushroom Kingdom into two sets: A and B, where each possible path to get to World 4-2 is in A and each possible path to get from World 4-2 to the end of the game is in B.

Here is a chart enumerating each path in A, where an “X” indicates a completed castle, $NC$ is the total number of completed castles for that path, and $NP$ is the total number of Princesses being in a castle for that path:

Path	World 1	World 2	World 3	NC	NP	Notes
$A_1$	X	X	X	3	0
$A_2$		X	X	2	0	Warp from 1-2 to 2-1
$A_3$			X	1	0	Warp from 1-2 to 3-1
$A_4$				0	0	Warp from 1-2 to 4-1

And a similar chart for set B:

Path	World 4	World 5	World 6	World 7	World 8	NC	NP	Notes
$B_1$	X	X	X	X	X	5	1
$B_2$		X	X	X	X	4	1	Warp from 4-2 to 5-1
$B_3$			X	X	X	3	1	Warp from 4-2 to 6-1
$B_4$				X	X	2	1	Warp from 4-2 to 7-1
$B_5$					X	1	1	Warp from 4-2 to 8-1

So the total number of paths possible is:

$\displaystyle \sum_{i, j} A_i+B_j=20$

(which really is just $A\times B=20$ )

To calculate the probability of The Princess being in another castle, $P(AnotherCastle)$ , we can calculate $P(Princess)$ and subtract it from 1 to give us:

$P(AnotherCastle) = P(\overline{Princess}) = 1-P(Princess)$

$P(Princess)$ is equal to the number of Princesses being in a castle in all possible paths, divided by the number of castles in all possible paths:

$\displaystyle P(Princess)=\frac{\sum_{i, j} NP(A_i+B_j)}{\sum_{i, j} NC(A_i+B_j)}=\dfrac{20}{90}=\dfrac{2}{9}$

(Note that $\sum_{i, j} NP(A_i+B_j)=20$ , since each possible total path $A+B$ results in 1 Princess being in a castle.)

So instead of $P(AnotherCastle)=\frac{7}{8}$ , the correct calculation is:

$P(AnotherCastle)=1-\dfrac{2}{9}=\dfrac{7}{9}$

Furthermore, since each path has a $\frac{1}{20}$ probability of being taken, we can calculate the expected values for the number of times the Princess is in another castle and the number of Princesses being in a castle:

$E(AnotherCastle)=\dfrac{70}{20}=3.5$

$E(Princess)=\dfrac{20}{20}=1$

To summarize: on an average full game of Super Mario Bros., one should expect:

The Princess to be in another castle about 78% of the time
The Princess to be in another castle 3.5 times
The Princess to be in a castle 1 time

We here at J. Cart. Overanal. feel this probabilistic description is far superior, assuming we didn’t screw up the math. Comments or complaints about wonky symbol useage are welcome below.

This entry was posted on Tuesday, March 24th, 2009 at 3:00 pm and is filed under Dr. Mario, Super Mario Bros., ex situ, philosophy. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to “Ex Sitibus: A Super Mario Triptych”

Ex Situ Redux: Study Shows The Princess is in Another Castle 7/8ths of the Time « The Journal of Cartoon Overanalyzations Says:
December 2, 2009 at 7:00 am | Reply
[...] Note: This is a republish of an old Ex Situ which was part of a set, but I thought with my addendum that it deserved its own [...]

The Journal of Cartoon Overanalyzations