Download presentation

Presentation is loading. Please wait.

Published byAnton Belvin Modified over 2 years ago

1
Conditional Probability

2
What is a conditional probability? It is the probability of an event in a subset of the sample space Example: Roll a die twice, win if total ≥ 9 Sample space S = set of outcomes = {11, 12, 13, 14, 15, 16, 21, 22, …, 65, 66} Event W = pairs that sum to ≥ 9 = {36, 45, 46, 54, 55, 56, 63, 64, 65, 66} Pr(W) = 10/36

3
What is a conditional probability? Now suppose we know that the first roll is 4 or 5. What is now the probability that the sum of the two rolls will be ≥ 9? Let B = first roll is 4 or 5 = {41, 42, …, 46, 51, 52, …, 56} Event W∩B = {45, 46, 54, 55, 56} Pr(W | B) = |W∩B|/|B| = 5/12 “Probability of W given B”

4
Conditional probability But since the sample space is the same, In general, the conditional probability of event A given event B is defined as

5
What is the difference between Pr(A|B) and Pr(B|A)? Pr(A|B) is the proportion of B that is also within A, that is, Pr(A|B) is |A∩B| as a proportion of |B| Pr(A|B) is close to 1 but Pr(B|A) is close to 0 AB A∩B

6
CS20 This class has 42 students, 13 freshmen, 17 women, and 5 women freshmen So if a student is selected at random, –Pr(Freshman) = 13/42, –Pr(Woman) = 17/42 –Pr(Woman freshman) = 5/42. If a random selection chooses a woman, what is the probability she is a freshman? –Simple way: #women freshmen/#women = 5/17 –Using probability:

7
Conditional Probability and Independence Fact: A and B are independent events iff Pr(A|B) = Pr(A). That is, knowing whether B is the case gives no information that would help determine the probability of A. Proof: A and B independent iff Pr(A)∙Pr(B) = Pr(A∩B) Pr(A∩B) = Pr(A|B)∙Pr(B) So as long as Pr(B) is nonzero, Pr(A)∙Pr(B) = Pr(A|B)∙Pr(B) iff Pr(A) = Pr(A|B)

8
Total Probability Suppose (hypothetically!): –Rick Santorum has a 5% probability of getting enough delegates to become the Republican nominee, unless the voting goes beyond the first ballot and there is a brokered convention –In a brokered convention, Santorum has a 65% probability of winning the nomination –There is a 7% probability of a brokered convention (cf. Intrade.com) What is the probability that Santorum will be the Republican nominee?

9
Total Probability Simple version: For any events A and B whose probability is neither 0 nor 1: That is, Pr(A) is the weighted average of the probability of A conditional on B happening, and the probability of A conditional on B not happening. B B _ A S

10
“Total probability” = weighted average of probabilities Pr(Santorum|Brokered) =.65 Pr(Santorum|¬Brokered) =.05 Pr(Brokered) =.07 Then Pr(Santorum) =.65∙.07 +.05∙.93 =.092

11
FINIS

Similar presentations

OK

Previous Lecture: Data types and Representations in Molecular Biology.

Previous Lecture: Data types and Representations in Molecular Biology.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on intellectual property act Free download ppt on resources and development Free ppt on rocks and minerals Seminar ppt on smart card technology Ppt on schottky diode model Ppt on design of earthen dam Ppt on obesity prevention program Differential display ppt online Ppt on introduction to production and operation management Best way to show ppt on ipad