1 1 Review Probability Axioms – Non-negativity P(A)≥0 – Additivity P(A U B) =P(A)+ P(B), if A and B are disjoint. – Normalization P(Ω)=1 Independence of two events A and B – P(A∩B)=P(A)P(B) – Two coin tosses A={first toss is a head}, B={second toss is a head} – Disjoint vs independent P(A∩B)=0, if P(A)>0 and P(B)>0, P(A∩B) < P(A)P(B). They are never independent. A={first toss is a head}, B={ first toss is a tail} P(A)=0.5, P(B)=0.5, P(A∩B)=0
14
Embed
0 0 Review Probability Axioms –Non-negativity P(A)≥0 –Additivity P(A U B) =P(A)+ P(B), if A and B are disjoint. –Normalization P(Ω)=1 Independence of two.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
1
Review
Probability Axioms – Non-negativity P(A)≥0
– Additivity P(A U B) =P(A)+ P(B), if A and B are disjoint.
– Normalization P(Ω)=1
Independence of two events A and B– P(A∩B)=P(A)P(B)
– Two coin tosses A={first toss is a head}, B={second toss is a head}
– Disjoint vs independent P(A∩B)=0, if P(A)>0 and P(B)>0, P(A∩B) < P(A)P(B). They are never
independent. A={first toss is a head}, B={first toss is a tail}
P(A)=0.5, P(B)=0.5, P(A∩B)=0
2
2
1.3 Conditional Probability 1.4 Total Probability Theorem and Bayes’
Rule
3
3
Conditional Probability
A way to reason about the outcome given partial information Example1
– To toss a fair coin 100 times, what’s the probability that the first toss was a head?
Fair coin 1/2
– To toss a fair coin 100 times, if 99 tails come up, what’s the probability that the first toss was a head?
Very small?
Example2– A fair coin and an unfair coin (1/4 tail, 3/4 head)
The first toss is fair, if the outcome is a head, use the fair coin for the 2nd toss, if the outcome is a tail, use the unfair coin for the 2nd toss.
What’s the probability that the 2nd toss was a tail? – ½x½ + ½x¼ = 0.375
What’s the probability that the 2nd toss was a tail if we know that the first toss was a tail?
– 1/4
4
4
Conditional Probability
Definition of a conditional probability– The probability of event A given event B (P(B)>0 )
– P(A|B)=P(A) if A and B are
independent
A new probability law (recall the definition of probability laws)–
5
5
Conditional Probability
Examples – Two rolls of a die, what’s the probability that the first roll was a 1?
– Fair dice 1/6
– Two rolls of a die, the sum of the two rolls is 6, what’s the probability that the first roll was a 1?
B: (1,5) (2,4) (3,3) (4,2) (5,1) , A and B: (1,5) P(A|B)= (1/36)/(5/36)=1/5
– Two rolls of a die, the sum of the two rolls is 6, what’s the probability that the first roll was EVEN?
B: (1,5) (2,4) (3,3) (4,2) (5,1) , A and B: (2,4) (4,2) P(A|B)= (2/36)/(5/36)=2/5
6
6
Conditional Probability
The new universe is B P(A1)> P(A2), does it mean that P(A1|B)> P(A2|B)? No!
– An Example: Two rolls of a die B: the sum of the two rolls is 4, (1,3) (2,2) (3,1) A1: the first roll was 1 or 2 A2: the first roll was 3, 4, 5 or 6