1. Definitions, Set Theory and Counting
Chapter 2: Probability
Definitions, Set Theory and Counting

2. Classical Definition
If a random event can occur in n equally likely and mutually exclusive ways, and if na of these ways have an attribute A, then the probability of occurrence of the event having attribute A is written as:

3. Relative Frequency Definition
If a random event occurs a large number of times n and the event has attribute A in na of these occurrences, then the probability of the event having attribute A is

5. S E1 E2 E4 E3 A

6. S E1 E2 E4 E3 A B

7. S E1 E2 E4 E3 A B

8. Independent Events vs. Mutually Exclusive Events
Independent events: If the occurrence or nonoccurrence of an event A, has no relation to or influence on the occurrence or nonoccurrence of an event B (and vice versa) then the two events are independent and
If A and B have no outcomes in common then A and B are mutually exclusive and

9. S E1 E2 A Ac E3 E4

10. Conditional Probabilities
If the probability of an event B depends on the occurrence of an event A, then we write prob(B|A), the probability of B given A has occurred OR the conditional probability of B given A has occurred.

11. A B

12. Conditional Probability of Independent Events

13. Total Probability Theorem
If B1, B2, …Bn represent a set of mutually exclusive and collectively exhaustive events, one can determine the probability of another event A from B1 B2 B3 B4 B5 B6 A S

14. Example: Total Probability Theorem
Experimentally we have found that in May the probability of the water temperature in the Navasota River reaching over 50°F when the river is flowing at a rate greater than 3000 cfs is The probability of the temperature being over 50°F when the river flowing at a rate less than 3000 cfs is The probability on any given day in May that the river will exceed 3000 cfs is What is the probability that the river will exceed 50°F on any random day in May?

15. Sampling Selecting r items from n items.
Can be done in one of two ways
With replacement
Without replacement
4 types of samples
Ordered with replacement
Ordered without replacement
Unordered with replacement
Unordered without replacement

16. Sampling-Ordered with replacement
The number of ways of selecting r items from n items with replacement and order is important is nr.
Example: 3 blocks labeled 1 to 3. How many ways can we draw 2 of the 3 blocks when order matters and the blocks are replaced each time?

17. Sampling: Ordered without replacement
r ordered items can be selected from n without replacement in n(n-1)(n-2)…(n-r+1) ways.
Commonly written as (n)r and called the number of permutations of n items taken r at a time.

18. Sampling without replacement, unordered (i.e. order does not matter).
Similar to ordered sampling without replacement except that in ordered sampling the r items selected can be arranged in r! ways. Thus r! of the ordered samples contain the same elements.
The number of different unordered samples will then be commonly written as
and called the binomial coefficient Gives the number of combinations of
selecting r items from n without replacement.

19. Unordered sampling with replacement
Selecting r items from n with replacement when order doesn’t matter is equivalent to selecting r items from n+r-1 items without replacement.
We can consider the population as containing r-1 more items than it really does since the items selected will be replaced.

20. Example: Sampling
Shortly after being put into service some buses manufactured by a certain company have developed cracks on the underside of the main frame. Suppose that a particular city has 20 of these buses, and cracks have actually appeared in eight of them.
How many ways are there to select a sample of five buses from the 20 for a thorough inspection?
In how many ways can a sample of five buses contain exactly five with visible cracks?
If a sample of five buses is chosen at random, what is the probability that at least four of the five has visible cracks?