Discrete Structures Probabilities

, Probability
Definitions :




·
Experiment :
thing trying to model


ex : Coin flip , rolling dice

·
Outcome : result of experiment

·
Sample space : Set of all possible outcomes

·
Distribution : Probability of each outcome in distribution

·
Random variable : represents something in sample space ( ?)
·
Uniform distribution : all outcomes have equal probability
↳
ex :
rolling any number on dice is 'n probability
·
Event : subset of sample space

↳
ex : roll an even number on 6 sided dice (" + n
+n)

·

Expected Value : "average" outcome of random variable
&
ex : Probability Winning .0
2 . 5 + 0 .

0 .
5 : $1 expected

0 .
5 $2
0 .
5 $0



Variance how far
·

: outcome is from expected value

& ex : Probability Winning x -



E[X]
0 .
5 $2 $1(2 1) -




0 .
5 $0 -

$(0 1) -




Var(X) [ (X-E(3) E(x2) (E(X])
↳
formula
-


: = or




ex :
E(2] = 0 .
5 .


(2) +
0 .



5-10 =
2
(t[x]) = 12 = /

2 -
1 = 1



·
Standard deviation : Square root of variance

random variables don't affect eachother
·
Independence : Two


Set-up : Pr[X =
x] = 40 %
↓
↓ probability of outcome
random
variable
Capital) outcome
Clowercase)