University of Helsinki Probability The likelihood of something - PowerPoint PPT Presentation

Zach Laster University of Helsinki

 Probability – The likelihood of something happening  Statistics – Models for predicting events based on previous occurrences  Using these, we can estimate what will happen or how things will progress in a game  We can also use them to balance probabilistic events, such as hit frequencies and gambling mini-games

 Probabilities are not guesses  Rolling a d6 results in a 16.7% percent chance of getting a 1. This is a fact. ○ Unless of course the die isn’t fair, but you get the point.  Throwing a fair coin has a 50% chance to land on either face.  Probabilities are facts . Things we know to be true.  We just use them to make guesses

Independent & Related Events  An independent event happens the same way every time regardless of how previous results went  Flip a coin again. Did the first one affect the second?  How about a die? ;)  A related event affect the probability of later events  Drawing a card from a deck obviously reduces the chances of drawing that card from the deck  Standard example: I have a bag of red and blue marbles…

Conditional Probability  Probabilities can be multiplied together to find the chance of the events happening together  Getting two heads in a row = ½ * ½  This allows us to chain events  We can also do this for related events  Chance of drawing two Queens from a deck (without putting the first one back) ○ = 4/52 * 3/51

 This multiplicative effect on decimal (or rational) numbers obviously results in smaller and smaller chances  We can improve our odds slightly by covering a larger range  Chance of drawing a heart from a deck = 13/52 = 1/4  Chance of drawing 4 hearts in a row ○ 13/52 * 12/51 * 11/50 * 10/49 = ~20%  Chance of rolling something higher than a 2 on a d6 = 4/6

In Reverse  Sometimes, calculating the chances of something happening is tricky  In these cases we can calculate the chance it won’t happen, and subtract that from 100%  This gives us the probability it WILL happen. Magic!  As a trivial example, what are the odds you will roll something other than a 6. Clearly this is the same as not rolling a 6, so we can just take the odds of rolling a 6 (1/6) and subtract that from 1  1- 1/6 = 5/6

 So what are the odds of throwing a 6 in six throws of a die?  Obviously, not 100%  This is kind of unintuitive, but search your feelings, you know it to be true  Actually, the easiest was to figure this out is to do solve the reverse  What are the odds we won’t throw a 6 in six throws?

 The odds of not throwing a 6: 5/6  The odds of not throwing a 6 six times in a row: 5/6 * 5/6 * 5/6 * 5/6 *5/6 * 5/6 = 33%  The odds of throwing a six, then, is 100% - 33% = 67%

Statistics  Statistics is a mathematical science pertaining to the collection, analysis, interpretation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities; it is also used for making informed decisions in all areas of business and government. – Wikipedia.org  Statistics is a mathematical science that deals with collecting and analyzing data in order to determine past trends, forecast future results, and gain a level of confidence about stuff that we want to know more about. – 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

 Statistics is a weird math science thing that can really get confusing  However, it is more useful than it has any right to be!  Statistics is probably something you are more familiar with than you realize

 A Population is the entire collection of everything we want to know something about  All the people online, all the people who play a kind of game, all the people in Finland/Helsinki  A Sample is a subset (AH! Math!) of the population. We use this to gather data and then make conclusions about the population at large.  We don’t perform the test on the entire population because, seriously, you want to ask EVERYONE on the internet/in Helsinki a dozen questions?

 Ideally, our sample size will be large. The closer it is to the population size the better.  If you have a population of 10,000 and you ask two people something, how well do you think that covered the entire population?  Of course, time and money simply don’t allow us to poll every person ever, so we use samples  In digital games, we can actually embed the polling into the game, so it automatically collects the data from every player! That’s actually a really amazing thing!

Distributions  Statistics has this nice tendency of producing similar distributions  This feels like there’s a joke in here somewhere  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.

The Normal Distribution  Also called the “Bell Curve” and the “Gaussian Distribution”  Here the population is closely centered around the mean or average value.

 In addition to being focused on a mean, the standard deviation and variance of a distribution are also worth note  Standard deviation is basically how far off the norm values are on average  Some things will be further out, others will be closer  An average of 3 minutes in a level with a standard deviation ( σ ) of 30 seconds is pretty good ○ On average, you’ll take from 2.5 minutes to 3.5 minutes to complete the level, with a tendency towards 3 minutes.

Margin of Error  If our population size is bigger than our sample size, then we have some margin of error  How far off we might be, given we didn’t include every element in the population  One method of this is a confidence interval, such as 95% certainty that something will hold true  Generally, “we can guarantee with A% confidence that B% of the data will be between values C and D.” (Sigman Part 2)  In statistics, more data is king. Always and forever, more data is better.

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”  Humans are terrible (and I mean terrible) at probability.  Really, we’re crap at it.  This leads to common misconceptions like “I just rolled three 1s! Clearly the next roll won’t be a 1.”  Or “I’ve not rolled a 20 in a while. I’m due.”  No matter what has happened in the past, the probability of rolling a 20 has not changed.  I don’t care if you haven’t rolled one tonight. Or this week. Or even this month. It’s still 1/20.

 Most gamers actually KNOW that probability doesn’t work that way  Smarter than your average gambler, we  Despite this, they still commit it frequently  “Dude! My dice are hot tonight!”  “Man, you stole one of my 20s!”  We’re that bad at probability.

Double Rares  Related to this is shock (and consternation) when someone gets two rares in a row (particularly, someone else)  For instance, I draw two treasure items, both of them are rare items  Lots of players will be really happy at their new windfall  The players around them may not be so happy, and feel the system is broken  “It just handed out two rares at once! That’s like two 1% chances in a row!”  Actually, if we think about this, obviously this SHOULD happen  1% * 1% = 0.01%, which isn’t likely, but it IS possible.

The Anti-law of Averages  The next standard error is that the number of rolls will average.  If we think about this one, it’s silly, but it still comes up  If we flip a coin 10 times and get an uneven split (which is actually kind of likely), it’s not reasonable to believe that throwing the coin 10 more times will make the numbers balance out.  This is because probability is a percentage

 Say we got an 8:2 split  If we throw it 10 more times, we could actually get the same split!  Statistically speaking, if we throw the coin a million times, we’d expect the split to be about 50%  However, the actual number of heads vs tails could be off by a huge amount.  In the long run, the difference in heads and tails flips will probably actually grow, not shrink

Selection Bias  This one actually comes from the fact that humans aren’t really good at recall, either  CogSci will actually tell us that we forget things because we couldn’t function otherwise ○ There are people who don’t, and they can’t  We actually forget bad things by design ○ So we don’t live in perpetual fear of door jams