Transcript MATH 3033 based on

MATH 3033 based on
Dekking et al. A Modern Introduction to Probability and Statistics. 2007
Slides by James Connolly
Format by Tim Birbeck
Instructor Longin Jan Latecki
C3: Conditional Probability And Independence
3.1 – Conditional Probability


Conditional Probability: the probability that an event
will occur, given that another event has occurred that
changes the likelihood of the event
Example:
If event L is “person was born in a long month”, and event
R is “person was born in a month with the letter ‘R’ in it”,
then P(R) is affected by whether or not L has occurred.
The probability that R will happen, given that L has
already happened is written as:
P(R|L)
  {Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec}.
3.1 – Conditional Probability
P( A  C )
P( A | C ) 
P(C )
Provided P(C) > 0
3.2 – Multiplication Rule
For any events A and C:
P( A  C )  P( A | C ) P(C )
3.3 – Total Probability & Bayes Rule
The Law of Total Probability
Suppose C1, C2, … ,CM are disjoint events such that
C1 U C2 U … U CM = Ω. The probability of an arbitrary
event A can be expressed as:
P( A)  P( A | C1) P(C1)  P( A | C 2) P(C 2)  ...  P( A | CM ) P(CM )
3.3 – Total Probability & Bayes Rule
Bayes Rule:
Suppose the events C1, C2, … CM are disjoint
and C1 U C2 U … U CM = Ω. The conditional probability of Ci,
given an arbitrary event A, can be expressed as:
P( A | Ci )
P(Ci | A) 
P( A)
or
P( A | Ci )
P(Ci | A) 
P( A | C1) P(C1)  P( A | C 2) P(C 2)  ...  P( A | Cm) P(Cm)
3.4 – Independence
Definition:
An event A is called independent of B if:
P( A | B)  P( A)
That is to say that A is independent of B if the probability of A occurring is not
changed by whether or not B occurs.
3.4 – Independence
Tests for Independence
To show that A and B are independent we have to prove just one of the following:
P( A | B)  P( A)
P( B | A)  P( B)
P( A  B)  P( A) P( B)
A and/or B can both be replaced by their complement.
3.4 – Independence
Independence of Two or More Events
Events A1, A2, …, Am are called independent if:
P( A1  A2  ...  Am)  P( A1) P( A2)...P( Am)
This statement holds true if any event or events is/are replaced by their
complement throughout the equation.