Discrete Math Notes:
Chapter 4: Counting and Discrete Probability
4.1 Conditional probability and independence
conditional probability, the second event F is that the blue die comes up 5. If the event F happens, the new probability of E
independent, two events are considered to be independent if conditioning on one event does not change the probability of the other event
mutually independent, Events At, -.., A, in a sample space S are considered to be mutually independent if the probability of the intersection of any subset of the events is equal to the product of the probabilities of the events in the subset
overview:
Sample Space S
10
S={a,b,c,d,e,f,g,h,i,j,k,I,m} IS| = 13
E
Event F={b,e,d,f.g} Event E={b,c,d,e}
0=(|w)d=(li)d=(ly)d =(l?d=(l!)d=(lu)d=(lo)d=(le)d :suaddeu JI
IF
Captions ^
1.The sample space S={a, b,c,d,e,f,g, hi,j,k, I,m}. ISI =13.The probability of each
for every event x not in F.
F
The first is the original event E, which states that the sum of the two integers must be at least 11. The blue die comes up 5 as the second event F. If the event F occurs, the new probability of E is denoted by p(E|F), which is the conditional probability of E given F. Conditioning on one event does not influence the probability of the other happening, hence they are independent. Mutual independence occurs only when the chance of any subset of events intersecting is equal to the product of the probabilities of the events in the subset.
4.2 Bayes' Theorem
Overview:
Bayes' Theorem in Action:
F : fair die selected F : loaded die selected X : the die comes up to 6
p(F)=p(F)=2 p(X|F)=1 p(X|F)=2
Bayes' Theorem:
p(X|F)p(F
p(F|X)= p(X|F)p(F)+p(X|F)p(F) 1.1 ~.37
Captions ^
1. F: fair die selected. F: loaded die selected.X: the die comes up 6. p(F) = p(F) = pX|F)=pX|F)=2 2. Bayes' Theorem: p(X|F)p(F) . Replace each occurrence of p(F) and p(F) p(X|F):p(F)+p(X|F).p(F) with . 3. Replace both occurrences of p(X|F) with 1/6)1/2)
The Bayes' Theorem is the foundation of algorithms that aim to calculate the probability of an event based on data collected from observations. Based on the evidence of the outcomes, it provides a technique to reason quantitatively about the likelihood that the die is loaded. The gambler, for example keeps two dice in her pocket and chooses one at random for a game. Is it possible to deduce which die she chose based on the results of the dice rolls? If a 6 appears more frequently than expected, it's likely that the gambler is using the loaded die. We'd like to know the probability of F or F if we see a 6. (i.e. conditioned on the event X). As a result, we need to know p(F(X). The Bayes' Theorem can be used to caIculate p(F|X) from p(X|F), p(X|F),and p(F)
4.3 Random variables
random variable, a function from the sample space S of an experiment to the real numbers. X(S) denotes the range of the function X
distribution, a random variable is the set of all pai