Probability and Conditional Probability: An Example from Consumer Complaints - Prof. Lei C, Study notes of Data Analysis & Statistical Methods

Download Probability and Conditional Probability: An Example from Consumer Complaints - Prof. Lei C and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity!

Chapter 3 (sections 5-8)

Probability

Conditional Probability

Conditional Probability – the probability that

event A occurs given that event B occurs

Conditional probability works with a reduced

sample space, the space that contains B

and

   

P A B

P A B P B

P B

A  B

Event A = {cause of complaint is appearance}

Event B = {complaint occurred during guarantee period}

Distribution of Product Complaints Reason for Complaint Complaint Origin Electrical Mechanical Appearance Totals During Guarantee Period 18% 13% 32% 63% After Guarantee Period 12% 22% 3% 37% Totals 30% 35% 35% 100% P  A  B  . 32 ^ ^    

. 51 . 63 . 32    P B P A B P A B 

Text, pp. 158, Example 3.

The Multiplicative Rule and Independent Events

The Multiplicative Rule

P  A  B   P  A  P  B A  or P  A  B   P  B  P  AB  P A   B  C (^)  ?

Solution 1: Example: Assignment #2, Q# ( S 1, S 2, S 3) ( S 1, S 3, S 4) ( S 1, S 4, S 5) ( S 1, S 5, F 1) ( S 1, S 2, S 4) ( S 1, S 3, S 5) ( S 1, S 4, F 1) ( S 1, S 5, F 2) ( S 1, S 2, S 5) ( S 1, S 3, F 1) ( S 1, S 4, F 2) ( S 1, S 2, F 1) ( S 1, S 3, F 2) ( S 1, F 1, F 2) ( S 1, S 2, F 2) ( S 2, S 3, S 4) ( S 2, S 4, S 5) ( S 2, S 5, F 1) ( S 2, F 1, F 2) ( S 2, S 3, S 5) ( S 2, S 4, F 1) ( S 2, S 5, F 2) ( S 2, S 3, F 1) ( S 2, S 4, F 2) ( S 2, S 3, F 2) ( S 3, S 4, S 5) ( S 3, S 5, F 1) ( S 3, F 1, F 2) ( S 3, S 4, F 1) ( S 3, S 5, F 2) ( S 3, S 4, F 2) ( S 4, S 5, F 1) ( S 5, F 1, F 2) ( S 4, S 5, F 2) ( S 4, F 1, F 2) a. P(selecting none F) = 10/ b. P(selecting one F) = 20/ c. P(selecting two F) = 5/

Solution 2: Example: Assignment #2, Q# (^5) 5! (^3) 3!(5 3)! 10 a. P(selecting none F)= 7 7! 35 3 3!(7^ 3)! (^5 2) 5! 2 (^2 1) 2!(5 2)! 20 b. P(selecting one F)= 7 7! 35 3 3!(7^ 3)! (^5 2) 5! 1 (^1 2) 1!(5 1)! c. P(selecting two F)= 7 7! 3                                     (^)                         5 35 3!(7 3)!  

The Multiplicative Rule and

Independent Events

Events A and B are independent if the occurrence of one does not alter the probability of the other occurring If A and B are independent events

P  A B   P  A  and P  B A   P  B 

P  A  B   P  A   P  B 

Event A = {cause of complaint is appearance} Event B = {complaint occurred during guarantee period} Are A and B independent events? A and B are not independent Distribution of Product Complaints Reason for Complaint Complaint Origin Electrical Mechanical Appearance Totals During Guarantee Period 18% 13% 32% 63% After Guarantee Period 12% 22% 3% 37% Totals 30% 35% 35% 100% P  AB  . 51 P  A  . 32 . 03 . 35

Text, pp. 158, Example 3.

Bayes’s Theorem (Optional)

Allows calculation of unknown conditional

probability from known conditional

probability

Read Text, pp. 174-175, Example 3.23.

                      k k i i i i

P B P AB P B P AB P B P A B

P B P A B

P B A P B A P A

1 1 2 2

Random Sampling

Assume a desired sample size of n

Sample is random if every set of n elements

in the population has the same probability of

being selected

Random number generators often used to

produce a random sample.

Read Text, pp. 171-172, Example 3.22.