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Probability theory is the branch of mathematics concerned with guessing and hand-waving and magic. The central objects of probability theory are random variables, random processes, and random shit that just happens. That's just life, man.

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History

The mathematical theory of probability has its roots in gambling, as most mathematicians are completely unable to meet the bills otherwise (see: don't judge). Blaise 'the Pimp' Pascal very was well known in the seventeenth century for solving many problems of the Probability (or probabilitation-tion, as it was known at the time) such as the "problem of points", "the problem of walking across bridges", "the problem of points walking across bridges", "the problem of pointy points" and the "problem of non-pointy potentiation points". Points. His arch-nemeses were Christiaan Huguenot and Laplace, which he vanquished in an epic dance-off.

Initially, probability theory was mainly considered discrete (i.e., how many shots I can polish off in one night), and its methods (drugs) were mainly (booze) combinatorial (booze and drugs). Eventually however, analytical considerations (i.e., the Mafia) compelled (broke my legs) the incorporation of continuous (how much of a never-ending bottle of vodka I can polish off in night) variables (vodka) into the theory.

The introduction of variables greatly attracted the Russian, who promptly perfected a version of probability minus the bullshit (see: Measure Theory). However the bullshit version of Probability theory is still widely taught in the United States, because here in America we don't listen to Communists (see: FREEDOM).

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Treatment

If one comes down with Probability theory, one should immediately consult a doctor. Unfortunately, it is lethal. Just shoot yourself .

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Motivation

Come on! You can do it! Go! Go! Go!

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Discrete probability distributions

The first thing you should know about Probability is that there are these things called dice (see: gambling). A die is a magical six-sided object, such that whenever you roll it, the number you want to come up will NEVER FUCKING COME UP (see: dirty fucking cheaters). Therefore, you can control the number that will come up by thinking really hard about the number you don't want to come up. This is called an Event.The person that thinks the hardest wins the Event. For this reason, Probability is one epic mind duel (see: Card Games on Motorcycles).

Another example of a discrete probability distribution is a random walk, which is sort of like what happens when you stumble home drunk from the bar one night and end up in North Korea instead (help).

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How to do the solving of the discrete probability

"1,009,853,101,042." – College freshman on the beach

Discrete probability is the by far the favorite topic of math professors, as all its issues can be solved by counting. It does not matter what the problem says. It does not matter what the textbook says. It does not matter what your professor says. It does not matter what your preconceived notions of Maths are. All your intuition is wrong. You must begin anew, young grasshopper.

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The Conditional Probability

The Condition Probability is a sub-species of the Probability. It is a rare creature, rapidly going extinct because of poaching and habitat loss. Only 5000 are estimated to exist in the world today. Attempts at breeding the Conditional Probability in captivity have proven tragically unsuccessful (see: Detroit). You must take this chance to observe it in the wild while you still can, young grasshopper.

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On the computing of the Conditional Probability

The Conditional Probability is always 0.312 except when it is 0.567, and then it is 0.222. The Conditional Probability is always between 0 and 1, except when it is below 0 or above 1 and then it is 0.5. The Conditional Probability is never equal to 0.793. NEVER. It is a well-known phenomenon of Quantum Physics that computing the Conditional Probability changes the value of the Conditional Probability, except when it is 0.421, and then it is 0.111.

Any Conditional Probability can easily be computed in three steps as follows: 1. Draw a circle. 2. Go to the bar and drink until you pass out.

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The Monty Hall Problem

Most Maths Professors like to start off a Probability Course by lecturing about the Monty Hall Problem for about one or two or three or four or five or fourteen lectures. The main point of the Monty Hall problem is that you DON'T switch doors because this is AMERICA and we DON'T MAKE MISTAKES here. (Note: in Russia, they switch doors. In Soviet Russia, door switch you!)

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The Monty Python Problem

The Monty Python Problem is different than the Monty Hall Problem. The Monty Python Problem is usually solved by a swift, forceful punch to the face (fish-slapping and head-exploding are also acceptable). Occasionally you may have to use a straightedge or compass to solve the Monty Python Problem.

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The Laws

In Probability there are many laws. Laws are things you must follow or you will be arrested and handcuffed and maybe tasered. At least that is what my parole officer tells me. If you are breaking the laws while solving the problems of Probability you are probably doing it wrong. Try turning the paper over.

These are the Laws you must be following while solving the Probability:

Law of Large Numbers: The First Law of Probability is the Law of Large Numbers which says that numbers are large. Next!

The Second Law: The Second Law of Probability is that we don't talk about Probability.

The Third Law: There is no Third Law.

The Weak Law: If you cannot toss a coin one-thousand times without collapsing you are weak. WEAK, SOLDIER! WEAK!

The Strong Law: STRONG LAW ALL IN CAPS BECAUSE STRONG LAW STRONG! NOT WEAK! NOT PUNY! THE STRONG LAW SAYS THE SAMPLE AVERAGE CONVERGES ALMOST SURELY TO THE EXPECTED VALUE OF THE AVERAGE AS N GOES TO INFINITY! PROOF OF STRONG LAW VERY COMPLEX! NOT GOING TO GO OVER HERE! TOO COMPLEX FOR YOUR PUNY BRAIN!

The Fourth Law: Almost surely you can't be serious.

The Fifth Law: I am. And don't call me Shirley.

Central Limit Theorem (CLT): C-C-C-COMBO BREAKER. The Central Limit Theorem is one of the greatest results of Mathematics. It explains the prevalence of the normal distribution in nature and the secret and the meaning and the purpose to life. The Central Limit Theorem is often abbreviated CLT. The Central Limit Theorem likes to refer to himself in the third person.

Badass Limit Theorem (BLT): The Badass Limit Theorem says we takin' things up to the limit up in here! *shoots off guns*

Bayes's Law: Write down an equation. Multiply out the denominators. Thank you. Thank you. I'm here all week.

Beyoncé's Law: To the left, to the left, everything you compute in a box to the left.

John Venn's Law: Fellow of the Royal Society John Venn says: if you cannot solve the Probability by counting, then draw two circles. Make sure they overlap. Shade in the overlapping part. This is called a Venn Diagram. Drink some tea (see: Like a Sir). If problem not solved, shade in other parts of Venn Diagram. Stare at Venn Diagram. If problem still not solved, draw more circles and cry.

Euler's Law: All the above laws were actually discovered by Euler first.

Fibonacci's Law: If you put two rabbits together they make more rabbits! Lots more rabbits! They go: 1,1,2,3,5,8,13,21… this has something to do with Probability… wait for it…

Mendel's Law: Peas.

Erdős's Law: Erdős is back. He's here to solve Math problems and kick ass. And he's all out of Math problems.

Oiler Law: If your Probability is not being solved by these laws, drench it in oil and set it on fire. Fire is beautiful.

Gauss's Law: No, no, you go away. We no like you.

Newton's Law: Which one? *tosses fabulous hair*

Godwin's Law: Nazis. Sorry.

CATS Law: All your base are belong to us. You have no chance to survive make your time.

Total Law of Probability: All of these laws sum to the TOTAL LAW OF PROBABILITY!

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Probability distributions

Many Probabilities are being awarded in the world today. They come in gold, silver, bronze, chicken and waffles, and peanut. They go like this:

Distributions of Probability by distribution:

Discrete Uniform 1411 Probabilities
Bernoulli 8034 Probabilities
Binomial 2 Probabilities
Negative Binomial -2 Probabilities
Poisson 764 Fish
Geometric 733 Triangles
Continuous Uniform 222 Medals
Gorgeous Uniform 507 Medals
Normal God, you guys are so weird
Gauss
9000 Probabilities
Newton 9001 Probabilities
Exponential 848 Probabilities
Gamma 10 Probabilities
Beta 767 Probabilities

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Convergence of random variables

Is this thing on? Thank God. I don't have much time left. I think they got Mary and Sue and Peter and Shirley. I'm the only one left. I'm the only one. I've got to get the message out. Anyone—if anyone out there is listening—please, for the love of God—what was that sound? Oh fuck no. No. They're coming. No, they're getting in through the door—they're—oh God. You've got to help—help me—please help—AHH AHHH AAAAAHHHH HELP THEY'RE CONVERGING! THE RANDOM VARIABLES! OH GOD! AAAAAAAAAAAAAAAH! AAAAH! HEEEELLLLLP! HEEEELLLP!

Revenge of the Disclaimer! : *static*