A small glance at how probability works

[drspyder]

YOU'RE WRONG. (why probability > you, and a small glance at how probability works)

- by drspyder


It angers me off to the point of violence when people say "duhr, my stunray didn't freeze 20 turns in a row, the freeze percentage must go up because all the formulas say so"

No. You're wrong. WRONG. *stabs*

It's called an independent event.

http://www.mathgoodies.com/lessons/vol6/...vents.html
http://mathforum.org/library/drmath/view/56494.html
http://whatismoskau.ytmnd.com/

An indepedent event is an event that does not depend on any other events, hence the name. Say that the Ice Scorchstone has a heal percentage of 25%, or 1/4. I know it doesn't. It's close enough. Shut up.

That means on turn one, if you use it, it has a 1/4 chance of activating.
On turn two, if you use it, it has a 1/4 chance of activating.
On turn three, if you use it, it has a 1/4 chance of activating.
On turn four, if you use it, it has a 1/4 chance of activating.
On turn five, if you use it, it has a 1/4 chance of activating.

PATTERN M I RITE?


"But... but... but... the charts say that it has a 42.24% chance for activating on turn two!"

or

"CHARTS SAY TURN ONE 24%, TURN TWO 42.24%, YOU LIE GO DIE LOLOL"


No. You're wrong. WRONG.

Here's how it works. The charts are right. It's just that you happened to have misinterpreted it them. Don't worry, most people make that mistake. What the chart means to say is that it has an X chance of activating on turn 1, a Y chance of activating on turn 1 or 2, a Z chance of activating on turn 1 or 2 or 3.

...and if you're the kind of person who doesn't need much proof, you can stop reading right there, unless you want to get confused with math, you like math, or you want to know why those formulas work.



Say that the Ice Scorchstone has a healing chance of 1/4. (1/4 is easier to work with than 24% or 6/25. Less confusion for you and me, and it's close enough.) So that means that on turn one, it has a, well, 1/4 chance of healing. Nothing too complicated yet, right?

Time for formulas.

The probability that the Ice Scorchstone heals on EITHER TURN ONE OR TWO can be represented by this expression:

[1 - (1 - 1/4)^2]
That simplifies to 7/16. What does that mean? That means that every sixteen battles that you decided to use the Icy Scorchstone on the first two turns, seven of the sixteen battles are expected to yield healage.

...and if you want to know where that formula comes from, see the bottom of this educational rant.


The probability that the Ice Scorchstone heals on EITHER TURN ONE, TWO , OR THREE can be represented by this expression:

[1 - (1 - 1/4)^3]
That simplifies to 37/64... see the above explanation for what the means.

..I'm going to say that you've noticed the pattern so far and supply you with a general formula.

[1 - (1 - P)^T]

P is the probability of the occurence. It is obviously less than or equal to 1, and greater than or equal to 0.

T is the number of turns that are given.

...and with that given, I hope that you, as the reader and THE CATALYST IN THIS WORLD OF STRIFE (o rly? ya rly. no wai!), will have a better understanding of probability, and why there are corpses nailed to the wall with "probability > you" scrawled underneath. Oh yes, I nearly forgot, here's the explanation for the first formula. Or, screw it, I might as well explain the whole formula as best as I can.


...don't ask any questions until I'm done explaining. Please.


P is the probability that the event happens, right?

Well, so 1 - P is the probability that the event doesn't happen. (If you're lost here, you might as well stop reading and go pet your dog.)

And (1 - P)^n is the probability that the event does not happen for n turns in a row, right? If you know why, skip on. If you don't, well... explaining something like that in concise terms is slightly beyond me. Sorry.

Well, the whole point of subtracting that from 1 to get the final expression is because (1 - P)^n shows the probability of the event NOT happening for n turns in a row, which is exactly the opposite of what you're looking for. Specifically, what you're looking for is every single possibly EXCEPT this one. Hence subtracting that expression from 1.


Have a nice day.


Contact information:

dogstew@gmail.com
Atheist Temple (AIM)
drspyder at the iDB

...no stealing info, kthx, if you must steal, do not plagiarize (word-for-word = no-no), and give me credit.



[edit] That misspelling in the title bugged me.

[othersideofthemirror]

...about time someone showed common sense! Just because you do something repeatedly does not change the chances of it happening.

Good information and clear examples, though you may want to ease up on the tone some, you're sounding really harsh.

[drspyder]

i'm a flaming troll.

no furry ideas plz

[Redwolves]

Go go Probability Man! ^_^

Nice job Dr. Spyder

Experimental Probability does not always equal theoretical probability kthnx.

[zmike]

Way to Go! I told you*! refering to old guild member who didn't believe me

[Ace94d]

10/10.

I'm not a math genius but I understood it!

[Dhuman]

I can flip a coin and it land on heads, but that doesn't mean the next one MUST be tails. I love it! A little common sense goes a long way. And I like the act of stabbing to get the point across.

[Q77]

Great article drspyder...not as revealing as most other articles, but not as dull either:

Quote: (If you're lost here, you might as well stop reading and go pet your dog.)

That's funny!!! A bit short but it shows I can study for my math exam by reading IDB articles...good job

[drspyder]

Oh man. I'm hesitant to bump this, but I remember back when I was active on iDB, and Christ I was insane. Prepubescent drspyder's mentally disturbed tone is uhh... strong enough to jump out and stab Future drspyder in the face until he remembers how to use probability again. I've long since quit playing Neopets, but if anybody wants me to continue getting math-related points off to you guys, I wouldn't mind doing so.

[joe10]

Actually, YOU'RE WRONG!, probability isn't that simple


Sub-Article:
IceSS doesn't freeze 21 times in a row?! It is more likely the 22th. A bigger glance.
(why you > probability)

Just thought people might want to understand some probability.

First we will start with some definitions.
probability - "the chance that a given event will occur"
"(mathematics) a number, between 0 and 1, expressing the precise likelihood of an event happening"

Now, we will assume for the following text that Ice Scorchstone freezes 25% of the time.

So, you start using the scorchstone on Round 1 and it doesn't freeze and you use it continuously until Round 21 and it still hasn't frozen.
The programmers of Neopets, however, made the chance of it freezing 25%.  According to the test so far, though, it has frozen 0/21 times.
In this situation, this actually happened and the first 21 times it did not freeze - there was no tampering at all.
The probability of the healing happening is 1 in 4.  If you were to continue the test (keeping in mind that probability is PRECISE), it WILL get to the ACTUAL probability of 1 in 4, even though it is currently 0 in 21.
For the probability to get to 1 in 4, you need an opposite situation in which you would heal almost 6 times in a row, to sort of "counter balance" this natural phenomenon.
That means that now, you do have a very small chance of the opposite thing happening (healing).
There you have it - you are more likely to heal the next round, no matter what you say.

[theonlysaneone]

joe10 Wrote:Actually, YOU'RE WRONG!, probability isn't that simple


Sub-Article:
IceSS doesn't freeze 21 times in a row?! It is more likely the 22th. A bigger glance.
(why you > probability)

Just thought people might want to understand some probability.

First we will start with some definitions.
probability - "the chance that a given event will occur"
"(mathematics) a number, between 0 and 1, expressing the precise likelihood of an event happening"

Now, we will assume for the following text that Ice Scorchstone freezes 25% of the time.

So, you start using the scorchstone on Round 1 and it doesn't freeze and you use it continuously until Round 21 and it still hasn't frozen.
The programmers of Neopets, however, made the chance of it freezing 25%. According to the test so far, though, it has frozen 0/21 times.
In this situation, this actually happened and the first 21 times it did not freeze - there was no tampering at all.
The probability of the healing happening is 1 in 4. If you were to continue the test (keeping in mind that probability is PRECISE), it WILL get to the ACTUAL probability of 1 in 4, even though it is currently 0 in 21.
For the probability to get to 1 in 4, you need an opposite situation in which you would heal almost 6 times in a row, to sort of "counter balance" this natural phenomenon.
That means that now, you do have a very small chance of the opposite thing happening (healing).
There you have it - you are more likely to heal the next round, no matter what you say.

No, YOU'RE WRONG. Each use of the Ice SS is an independent event. It doesn't matter if you had used it 21 times before and not gotten anything, because one use does not affect the next. This is what the entire article is about.

Joe, you're referring to the Law of Large Numbers. If you were to try 1000 times, you would probably get 250 successes and 750 failures. This is because, over a lot of tries, the wins and losses even out. There is nothing saying it HAS to happen this way, only that it tends to.

Think of it this way: If you flip a coin 5 times and don't get heads, are you "due" for a heads next turn?