Conditional Probability Independent & Related Events An - PDF document

8/6/2014 Zach Laster University of Helsinki  Probability – The likelihood of something  Probabilities are not guesses happening  Rolling a d6 results in a 16.7% percent  Statistics – Models for predicting events chance of getting a 1. This is a fact. ○ Unless of course the die isn’t fair, but you get based on previous occurrences the point.  Using these, we can estimate what will  Throwing a fair coin has a 50% chance to happen or how things will progress in a land on either face. game  Probabilities are facts . Things we know  We can also use them to balance to be true. probabilistic events, such as hit  We just use them to make guesses frequencies and gambling mini-games Conditional Probability Independent & Related Events  An independent event happens the same  Probabilities can be multiplied together way every time regardless of how previous to find the chance of the events results went happening together  Flip a coin again. Did the first one affect the second?  Getting two heads in a row = ½ * ½  How about a die? ;)  This allows us to chain events  A related event affect the probability of later events  We can also do this for related events  Drawing a card from a deck obviously reduces  Chance of drawing two Queens from a deck the chances of drawing that card from the deck (without putting the first one back)  Standard example: I have a bag of red and blue marbles… ○ = 4/52 * 3/51 1

8/6/2014 In Reverse  This multiplicative effect on decimal (or  Sometimes, calculating the chances of rational) numbers obviously results in something happening is tricky smaller and smaller chances  In these cases we can calculate the chance it won’t happen, and subtract that from 100%  We can improve our odds slightly by  This gives us the probability it WILL happen. Magic! covering a larger range  As a trivial example, what are the odds you will  Chance of drawing a heart from a deck = 13/52 roll something other than a 6. Clearly this is = 1/4 the same as not rolling a 6, so we can just  Chance of drawing 4 hearts in a row take the odds of rolling a 6 (1/6) and subtract ○ 13/52 * 12/51 * 11/50 * 10/49 = ~20% that from 1  Chance of rolling something higher than a 2 on  1- 1/6 = 5/6 a d6 = 4/6  So what are the odds of throwing a 6 in  The odds of not throwing a 6: 5/6 six throws of a die?  The odds of not throwing a 6 six times in  Obviously, not 100% a row: 5/6 * 5/6 * 5/6 * 5/6 *5/6 * 5/6 =  This is kind of unintuitive, but search your 33% feelings, you know it to be true  The odds of throwing a six, then, is  Actually, the easiest was to figure this 100% - 33% = 67% out is to do solve the reverse  What are the odds we won’t throw a 6 in six throws? Statistics  Statistics is a mathematical science pertaining to the  Statistics is a weird math science thing collection, analysis, interpretation, and presentation of data. It is applicable to a wide variety of academic that can really get confusing disciplines, from the physical and social sciences to the humanities; it is also used for making informed decisions  However, it is more useful than it has any in all areas of business and government. – Wikipedia.org right to be!  Statistics is a mathematical science that deals with collecting and analyzing data in order to determine past  Statistics is probably something you are trends, forecast future results, and gain a level of confidence about stuff that we want to know more about. more familiar with than you realize – Tyler Sigman  Statistics can help you shine a flashlight upon your broken mechanics and shattered design dreams. It does this by giving you actual hard, scientific data to support meaningful design decisions. – Tyler Sigman 2

8/6/2014  A Population is the entire collection of  Ideally, our sample size will be large. The everything we want to know something closer it is to the population size the better. about  If you have a population of 10,000 and you ask two people something, how well do you think  All the people online, all the people who play a that covered the entire population? kind of game, all the people in Finland/Helsinki  Of course, time and money simply don’t  A Sample is a subset (AH! Math!) of the allow us to poll every person ever, so we population. We use this to gather data and use samples then make conclusions about the population at large.  In digital games, we can actually embed the polling into the game, so it automatically collects  We don’t perform the test on the entire the data from every player! That’s actually a population because, seriously, you want to ask really amazing thing! EVERYONE on the internet/in Helsinki a dozen questions? Distributions The Normal Distribution  Statistics has this nice tendency of  Also called the “Bell Curve” and the “Gaussian Distribution” producing similar distributions  This feels like there’s a joke in here  Here the population is closely centered around the somewhere mean or average value.  A distribution is basically a pattern which statistical data follows  For instance, we tend to have a central value which is common, and as we deviate from this value the probability of the new value drops. Margin of Error  In addition to being focused on a mean, the  If our population size is bigger than our sample standard deviation and variance of a size, then we have some margin of error distribution are also worth note  How far off we might be, given we didn’t include  Standard deviation is basically how far off every element in the population the norm values are on average  One method of this is a confidence interval,  Some things will be further out, others will be such as 95% certainty that something will hold closer true  An average of 3 minutes in a level with a  Generally, “we can guarantee with A% confidence standard deviation ( σ ) of 30 seconds is pretty that B% of the data will be between values C and D.” good (Sigman Part 2) ○ On average, you’ll take from 2.5 minutes to 3.5  In statistics, more data is king. Always and minutes to complete the level, with a tendency towards 3 minutes. forever, more data is better. 3

8/6/2014 No such thing as certain  I tried to explain this to a lawyer once. It didn’t go well.  Basically, you can’t reach a point where you’ve tested every possible thing  This is why there will always be bugs in your code  This is why you can’t actually rule out your neighbor being an alien  But you can be reasonably certain (like >90%) and that’s usually good enough  Go on, live a little. Who needs to be sure? “Stop stealing my good rolls!”  Known as “The Gambler’s Fallacy”  Most gamers actually KNOW that  Humans are terrible (and I mean terrible) at probability doesn’t work that way probability.  Really, we’re crap at it.  Smarter than your average gambler, we  This leads to common misconceptions like “I  Despite this, they still commit it just rolled three 1s! Clearly the next roll won’t be a 1.” frequently  Or “I’ve not rolled a 20 in a while. I’m due.”  “Dude! My dice are hot tonight!”  No matter what has happened in the past, the  “Man, you stole one of my 20s!” probability of rolling a 20 has not changed.  I don’t care if you haven’t rolled one tonight. Or this  We’re that bad at probability. week. Or even this month. It’s still 1/20. Double Rares The Anti-law of Averages  Related to this is shock (and consternation) when  The next standard error is that the someone gets two rares in a row (particularly, number of rolls will average. someone else)  If we think about this one, it’s silly, but it still  For instance, I draw two treasure items, both of them are rare items comes up  Lots of players will be really happy at their new windfall  If we flip a coin 10 times and get an  The players around them may not be so happy, and uneven split (which is actually kind of feel the system is broken likely), it’s not reasonable to believe that  “It just handed out two rares at once! That’s like two 1% chances in a row!” throwing the coin 10 more times will  Actually, if we think about this, obviously this make the numbers balance out. SHOULD happen  1% * 1% = 0.01%, which isn’t likely, but it IS possible.  This is because probability is a percentage 4