Season 9 ~ Episode #9 - - - Toothpaste and Orange Juice Experiment

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K-ModPermalink Reply by Gumbo123 on January 21, 2016 at 2:38pm

I AM A DICE GENIUS!!!

Legitimately, what are the chances?  Somebody do the math!

Well, Rhett, according to http://www.math.hawaii.edu/~ramsey/Probability/TwoDice.html . . . simply put, rolling a seven is the most likely outcome with 6 out of 36 or 1/6 = 16.667% outcome probability. 

If you wanted to really be the DICE GENIUS you should have gone for snake eyes (2) or box cars (12) which each have only 1/36 or 2.778% chance of happening on any single roll of two dice.

When rolling two dice, distinguish between them in some way: a first one and second one, a left and a right, a red and a green, etc. Let (a,b) denote a possible outcome of rolling the two die, with a the number on the top of the first die and b the number on the top of the second die. Note that each of a and b can be any of the integers from 1 through 6. Here is a listing of all the joint possibilities for (a,b):

(1,1)	(1,2)	(1,3)	(1,4)	(1,5)	(1,6)
(2,1)	(2,2)	(2,3)	(2,4)	(2,5)	(2,6)
(3,1)	(3,2)	(3,3)	(3,4)	(3,5)	(3,6)
(4,1)	(4,2)	(4,3)	(4,4)	(4,5)	(4,6)
(5,1)	(5,2)	(5,3)	(5,4)	(5,5)	(5,6)
(6,1)	(6,2)	(6,3)	(6,4)	(6,5)	(6,6)

Note that there are 36 possibilities for (a,b). This total number of possibilities can be obtained from the multiplication principle: there are 6 possibilities for a, and for each outcome for a, there are 6 possibilities for b. So, the total number of joint outcomes (a,b) is 6 times 6 which is 36. The set of all possible outcomes for (a,b) is called the sample space of this probability experiment.

With the sample space now identified, formal probability theory requires that we identify the possible events. These are always subsets of the sample space, and must form a sigma-algebra. In an example such as this, where the sample space is finite because it has only 36 different outcomes, it is perhaps easiest to simply declare ALL subsets of the sample space to be possible events. That will be a sigma-algebra and avoids what might otherwise be an annoying technical difficulty. We make that declaration with this example of two dice.

With the above declaration, the outcomes where the sum of the two dice is equal to 5 form an event. If we call this event E, we have

E={(1,4),(2,3),(3,2),(4,1)}.

Note that we have listed all the ways a first die and second die add up to 5 when we look at their top faces.

Consider next the probability of E, P(E). Here we need more information. If the two dice are fair and independent , each possibility (a,b) is equally likely. Because there are 36 possibilities in all, and the sum of their probabilities must equal 1, each singleton event {(a,b)} is assigned probability equal to 1/36. Because E is composed of 4 such distinct singleton events, P(E)=4/36= 1/9.

In general, when the two dice are fair and independent, the probability of any event is the number of elements in the event divided by 36.

What if the dice aren't fair, or aren't independent of each other? Then each outcome {(a,b)} is assigned a probability (a number in [0,1]) whose sum over all 36 outcomes is equal to 1. These probabilities aren't all equal, and must be estimated by experiment or inferred from other hypotheses about how the dice are related and and how likely each number is on each of the dice. Then the probability of an event such as E is the sum of the probabilities of the singleton events {(a,b)} that make up E.

Permalink Reply by MothTurtle on January 23, 2016 at 2:41pm

My parents have the exact same metal cup (is it a beer stein?). 

K-ModPermalink Reply by Alexis (Indigo Sparrow) on February 10, 2016 at 8:37pm

I generally don't have much problem with the "toothpaste v. orange juice" dilemma mainly because I don't drink orange juice that often. Buuuuut, I really want to try that orange flavored toothpaste. I find traditional toothpaste flavors to be boring... (I'll pass on the bacon-paste, though)

The spitting into the same cup grossed me out quite a bit... not as much as the giant gum wad, but still... ;-;