1. Events and their probability
Basic relationships of probability
Conditional probability

2. Events and their probabilities
An event is a collection of sample points.
The probability of an event is equal to the sum of the probabilities of the sample points in the event.

3. If we can identify all the sample points in an event, and assign probabilities to each, we can compute the probability of an event.

4. Events and Their Probabilities
Event M = Markey Oil is Profitable
M = {(10, 8), (10, -2), (5, 8), (5, -2)}
P(M) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2) =
= .70

5. P(C) = P(10, 8) + P(5, 8) + P(0, 8) + P(-20, 8)
Events and Their Probabilities
Event C = Collins Mining Profitable
C = {(10, 8), (5, 8), (0, 8), (-20, 8)}
P(C) = P(10, 8) + P(5, 8) + P(0, 8) + P(-20, 8) =
= .48

6. Basic Relationships of Probability
We can use these rules to compute the probability of an even when we don’t know the probability of all the sample points.
Complement of an event
Union of two events
Intersection of 2 events
Mutually exclusive events

7. Complement of an Event AC Event A
The complement of event A (denoted by AC) includes all sample points outside of event A.
Complement of A
Event A AC
Venn diagram
Sample space

8. Computing Probability Using the Complement
We’re trying to get you to think of probability as area or space
Note that:
Thus:

9. Union of Events
The union of events A and B is the event containing all points in A or B or both.
The union of events A ad B is denoted by
Event A
Event B

10. M C = {(10, 8), (10, -2), (5, 8), (5, -2), (0, 8), (-20, 8)}
Union of Two Events
Event M = Markley Oil Profitable
Event C = Collins Mining Profitable
M C = Markley Oil Profitable
or Collins Mining Profitable
M C = {(10, 8), (10, -2), (5, 8), (5, -2), (0, 8), (-20, 8)}
P(M C) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2)
+ P(0, 8) + P(-20, 8) =
= .82

11. Intersection of 2 Events
The intersection of events A and B contains all sample points in both A and B.
The intersection of events A and B is denoted by
Sample space
Event A
Event B
Intersection of events A and B

12. Intersection of 2 Events
Event M = Markley Oil Profitable
Event C = Collins Mining Profitable
M C = Markley Oil Profitable
and Collins Mining Profitable
M C = {(10, 8), (5, 8)}
P(M C) = P(10, 8) + P(5, 8) =
= .36

13. Addition Law
This law provides a means of calculating the probability of event A or event B or both occur.

14. Notice we must subtract
or else we will count that area twice and overstate the probability of A or B occurring
Sample space
Event A
Event B
Intersection of events A and B

15. Addition Law Event M = Markley Oil Profitable
Event C = Collins Mining Profitable
M C = Markley Oil Profitable
or Collins Mining Profitable
We know: P(M) = .70, P(C) = .48, P(M C) = .36
Thus: P(M  C) = P(M) + P(C) - P(M  C) =
= .82
(This result is the same as that obtained earlier
using the definition of the probability of an event.)

16. Mutually Exclusive Events
Events A and B are said to be mutually exclusive if they have no sample points in common.
Events A and B are mutually exclusive if, when one event occurs, the other cannot occur
Event A
Event B
Note that:

17. Conditional Probability
If the probability of being promoted to detective is influenced by whether you are a man or woman, then probability is conditional

18. Conditional Probability
The notation P(A | B) reads “the probability of A given B.
Conditional probability is calculated by:

19. Conditional Probability
Event M = Markley Oil Profitable
Event C = Collins Mining Profitable
= Collins Mining Profitable
given Markley Oil Profitable
We know: P(M C) = .36, P(M) = .70
Thus:

20. Multiplication Law Or:
This allows us to compute the probability of the intersection of 2 events. That is, what is the probability that both A and B will occur?
Or:

21. Multiplication Law Event M = Markley Oil Profitable
Event C = Collins Mining Profitable
M C = Markley Oil Profitable
and Collins Mining Profitable
We know: P(M) = .70, P(C|M) = .5143
Thus: P(M  C) = P(M)P(M|C)
= (.70)(.5143)
= .36
(This result is the same as that obtained earlier
using the definition of the probability of an event.)

22. Independent Events That is: and:
Events A and B are said to be independent if the probability of event A is not changed by the existence of event B, and vice-versa.
That is:
and:

23. Multiplication Law for Independent Events
The multiplication law can be used to determine if two events are independent

24. Multiplication Law for Independent Events
Event M = Markley Oil Profitable
Event C = Collins Mining Profitable
Are events M and C independent?
DoesP(M  C) = P(M)P(C) ?
We know: P(M  C) = .36, P(M) = .70, P(C) = .48
But: P(M)P(C) = (.70)(.48) = .34, not .36
Hence: M and C are not independent.