General Probability Rules…

If events A and B are completely independent of each other (disjoint) then the probability of A or B happening is just:

( ) ( ) ( )P A B P A P B

( ) ( ) ( )P A B P A P B

( ) ( ) ( ) ( )P A B C P A P B P C

We can extend the rule

Probability Rules

If events A and B are independent of each other (but not disjoint) then the probability of A and B happeningA and B happening is just:

( ) ( ) ( )P A B P A P B

Probability Rules

If events A and B are independent of each other (but not disjoint) then the probability of A or B happeningA or B happening is just:

( ) ( ) ( ) ( ) ( )P A B P A P B P A P B

Hmmm – why is this “less” than the disjoint case?

Examples…

4.86 4.89 4.94

Conditional Probability

Sometimes, knowledge of an event alters the probability of a future event. Example 4.30 illustrates this.

We write this as P(B|A), which represents the probability of B the probability of B happening given the occurrence of Ahappening given the occurrence of A

Multiplication rules …

( ) ( ) ( | )

( )( | )

( )

P A and B P A P B A

P A and BP B A

P A

Examples…

4.1014.103

Tree Diagrams…

These are useful when there are a large number of probabilities to consider

Example: 38% of people earn a post secondary degree. What is the probability of selecting a person at random from a large crowd so that the person is either female and has a PhD or male with no post secondary degree? Use the data from 4.94 and assume 50% of the general population is male.

Plan of attack:

•Lay out all stems and branches

•Assign probabilities

•Calculate!

Decision Analysis…

This is a very useful application of stats!Applications:

Medicine course of treatment (example 4.37) Computing Science “fuzzy logic” and AI Engineering/Business production choices

What’s the best option?

In conclusion…

Make sure you understand what is meant by conditional probability

Learn how to use (rather than memorize!) the probability formulae

Ignore the sections on Baye’s RuleTry 4.91, 4.92,4.97