Conditional Probability Independent & Related Events An - - PDF document

8/6/2014 1

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

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 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

• ut 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

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 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.

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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

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 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

• therwise

○ 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  So what does this mean for our perspective on

probability?

 Well, clearly good things happen more often

○ At least, so our memory would tell us

 So do rare events, oddly enough  That’s a fun one  We know the event was rare, so it sticks in our mind

when it happens

 The fact that it sticks so well (makes such an

impression) actually makes us feel like it happens more often

 This is why people are afraid of flying and not

driving, even though you are much more likely to die in a car accident

 This also mean we think we are cooler/better/more

skilled than we actually are

 While, yes, some of us have a hard time letting go of that

• ne time in 5th grade where we did that stupid thing in

front of everyone, overall we tend to remember the good things better

 This means we remember winning, particularly winning

epically, more than we do losing  Players therefore think they are better than they are

 This can result in them overestimating their abilities and

getting into situations (higher tier games) they can’t win

 We can fix this for example by auto-matchmaking or

dynamic difficulty

Self-Serving Bias

 Players sometimes feel that they aren’t

winning as often as they think they should, given their odds.

 They will also feel an unlikely event is

much more likely than it is overall (which ties back to our bad memories)

 Thus they will complain about losing 25% of the

time when they have a 75% chance to win

 But they will be fine with winning 25% when they

• nly have a 25% chance to win.

Attribution Bias

 This is in some ways humans being kind of

immature, because we are

 If the game rewards the player randomly,

then it was something they did

 “Man, I made the right decision going this way.”  If it penalizes them randomly, it’s the

game’s fault

 “This game is so not fair!”  Or worse, another player’s fault!

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Anchoring

 Basically ,getting stuck on the first number  Big numbers mean big chances, right?  Slot machines with big jackpots  Chances presented in larger scales (2:1 vs 20:10)  Basically, people get caught up in the

number sounding or feeling big, and don’t realize it’s actually fairly small or the same

 A player with a small base number and lots

• f bonuses may also misjudge the full

import fo those bonuses, thinking they are weaker than they are.

The Hot-Hand Fallacy

 This is the idea that someone on a streak

is more likely to score/strike/win again

 It largely originates form Basketball  Players would do well, and we’d term them as being “on fire”  NBA Jam actually made this a mechanic, giving a player on fire bonuses to speed and such

○ I loved this game, just for this mechanic. It made

you feel pretty awesome.  It’s completely wrong, though  Probability theorists laughed it off and

figured this was sillyness

 Basketball fans pointed to some things like

morale and flow states  Probability theorists sat down with some

statistics to figure out if it held

 And everyone was wrong  Turns out, a player on a streak is more likely to fail next time than succeed  This increased fail chance gets bigger the longer the streak  We’re not actually sure what produces

this

 Maybe getting cocky  Maybe getting tired  Maybe getting distracted  But it seems to be mostly psychological  It also seems to affect most forms of

streak, at least when the player is aware

• f it

Problems?

 The players will also blame you for some of their

fallacies

 When the actual tendency for things to occur happens,

they will feel the game is off, and think it’s doing it wrong.  Sid Meier’s 2010 Keynote made this rather clear

 When told they have 2:1 odds, a player will actually

expect to win slightly more often than that

 When told they gave 20:10 odds, they will expect to win

EVEN MORE often

○ It’s the same odds! ○ Once more, we’re terrible at expectations for odds

 Players will actually expect the odds to play more in their

favor than what they are told.

Managing This

 Skew the odds  Adjust the odds internally so they ARE actually higher than the presented odds  Limit random impact  Reduce the power of random events  Don’t let a single roll screw up a player (too badly)  Show what caused the problem to the player and what could be done to prevent that  Downplay streaks  We can combat the Hot-Hand fallacy by downplaying the value of streaks  Give actual mechanics bonuses to streaks  Or we can make streaks actually give a positive feedback loop

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Ethical?

 Lots of people will (reasonably) have

issues with the ethics of the first one, there

 It’s probably wrong to endorse peoples’

misconceptions  There are some things we can bias without

being unethical though

 ex: We can bias random drops so that if a player

hasn’t seen a rare lately, they get one

○ If we’re up front about this, no one will complain

(the players will love us, actually) and we’re in the clear for ethics concerns!

A better solution

 Instead of biasing things or downplaying

random elements, we can provide the player with all the information about all the random rolls thus far

 A table of all the outcomes/rolls will help a lot  Actual calculated percentages will really set the

player at ease

○ When they can look and see that the system IS

being balanced and fair, they will typically accept that and feel much better about the game

Monty Hall and You

 To be fair, probability is kind of weird.  There was (is?) a game show called “Let’s make a

Deal” which included the so called “Monty Hall Problem”

 Which is actually named after the host of the show at the

time…  There are three doors. Behind two of the doors there

is a goat, and the third has a shiny new car (or other desirable object. High-end computer?)

 The player chooses as door.  The host then opens one of the doors not chosen by

the player to reveal a goat.

 The player is then given the chance to change doors.

Should they?  Most people go, “well, it should be

independent, right? Why would I change? I now have a 50% chance of being right.”

 Which is false  Actually, you should change. Why?

Because there’s a 66% chance it’s the

• ther door.

 Because probability is weird.  This isn’t intuitive. It’s not obvious. It fooled

Paul Erdős…

 This only becomes apparent when you look

at a computer simulation of it, really.

 In order to make this make sense, keep in

mind we are dealing with the probability from the start

 The probability doesn’t actually change at any

• point. If you consider it from this point NOW you

are actually doing it wrong.  Think of it from the start and if the host

didn’t actually OPEN the door.

 You actually have the option of staying with the

door you chose or taking BOTH of the other two doors.

 You know one of those has a goat anyway, but

there’s a 2/3rds chance the car’s in one of them

Statistics Gone Wrong

 It’s easy to simply throw out statements

and make invalid conclusions from them

 “Team A hasn’t won a game against Team B

since 1982”

○ So what’s the probability of Team A beating

Team B? Probably not a lot to do with that statement.

○ So while this trivia is interesting from a

sportscasting perspective, it’s not statistically relevant.

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 Why the margin of error matters  “50% of people have been attacked by a shark in the past week!”

○ If the MoE is around 50%, then clearly this isn’t a

worthwhile statement

○ If it’s closer to 2%, we’re in trouble, and I’m breaking

• ut the Jaws theme

 Discarding valid results because you didn’t like

the results

 Making models off of flawed (biased/fake) data  If your data doesn’t hold water, then neither will the results.  Asking a loaded question obviously biases data!  It’s also easy to infer relationships that aren’t

there

 If you find two trends, you can easily graph them and

say, “Hey, look, the number of pizzas ordered per day coincides with the number of personal computers bought since 2000”

○ Clearly, the pizzas aren’t buying computers nor the

• ther way around

○ “Correlation does not imply Causality!” ○ Caution: there may not be such a correlation, I made it

up (It just seemed reasonable)  However, there may be a relationship between the

two.

○ Chances are, the two things ARE linked, but only by the

average wealth of people or something like that.

Become a Pirate, Save the Environment!

Luck vs Skill

 Some games are 100% skill based  Chess, Go  Most deterministic games  The opposite is possible  Chutes and ladders or Candyland  These don’t actually correlate as simply as

we might think with casual games or fun

 Tic-Tac-Toe is deterministic, but not big on skill  Poker is highly random, but we think of it as

requiring a lot of skill

Poker vs Blackjack

 Blackjack is also a card game with

similar levels of randomness to Poker

 But it’s definitely not the same level of skill

game (unless you are counting cards)

 Poker gives you a lot more moves and

• ption than blackjack does, and allows

you to adapt to changes in the environment

 You have far fewer options in Blackjack

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Professional Sports

 Sports games aren’t really random things. You

win or lose based on how good you are.

 Thus we pay players a fixed amount based on their

skill  And yet we talk about how likely they are to

win and chances to score

 Gamblers bet on these games as if they were

games of chance. Why’s this?

 From their frame of reference, it is.  They cannot influence the odds or the outcome of

the game, but they can estimate it based on past events

Action Games

 Hidden miss chances are bad  If it looks like you will hit, you should  From this, we can actually extrapolate

that hidden negative chances will leave players unhappy

 However, chances for random positive

events are better

 Critical hits  Headshots (ok, not strictly random, but they

happen randomly and work similarly)

Balanced Odds

 We discussed loot drops as positive

rewards yesterday

 We’ve also mentioned how we can skew their

frequency  We’ve also commented on how chances

for negative events tend to end up

 One thing taken from D&D: Randomness

favors the underdog

 The big dog could win without it, the underdog

has a larger chance of winning simply because they might get lucky

○ Two-face is wrong!

Value of something with a random chance

 We also need to balance our items and

mechanics with non-deterministic elements so that they are fair

 What are appropriate cost/benefit values for

something with a random chance?

 We’ll actually cover this at the beginning

• f tomorrow.

 Computers don’t really do random  Unless you have something really fancy built into it,

• f course, like something radioactive that decays at

random intervals…  Instead, computers use formulas to produce

numbers in a sequence

 Wait! How is that at all random?  It isn’t  The catch is that the sequence is built from a

starting number, which is kind of randomly selected

 Common methods are to ask the user or use the

current time

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 When designing a game where random is

important, and there is a lot of random (online poker) we need to get more fancy

 Especially in games with money riding on them.

People don’t like it if there are patterns in your random number generators. That can give people a way to cheat!  This means we need more entropy in our

• riginal seeds, and probably need to

change our seed more often

 Most games set the seed once and never

change it, but most games aren’t online poker

Homework

 Consider if your game is dominated

more by luck or skill

 Is this appropriate?  How's the perception of random in your

game?

 How’s the random in actuality?