1. Probability

2. Randomness
Long-Run (limiting) behavior of a chance (non-deterministic) process
Relative Frequency: Fraction of time a particular outcome occurs
Some cases the structure is known (e.g. tossing a coin, rolling a dice)
Often structure is unknown and process must be simulated
Probability of an event E (where n is number of trials) :

3. Set Notation S  a set of interest Subset: A  S  A is contained in S
Union: A  B  Set of all elements in A or B or both
Intersection: AB  Set of all elements in both A and B
Complement: Ā  Set of all elements not in A

4. Probability
Sample Space (S)- Set of all possible outcomes of a random experiment. Mutually exclusive and exhaustive form.
Event- Any subset of a sample space
Probability of an event A (P(A)):
P(A) ≥ 0
P(S) = 1
If A1,A2,... is a sequence of mutually exclusive events (AiAj = ):

5. Counting Rules for Probability (I)
Multiplicative Rule- Brute-force method of defining number of elements in S
Experiment consists of k stages
The jth stage has nj possible outcomes
Total number of outcomes = n1...nk
Tabular structure for k = 2
Stage 1\ Stage2 1 2 … n2
n2+1
n2+2
2n2 n1
(n1-1)n2+1
(n1-1)n2+2
n1n2

6. Counting Rules for Probability (II)
Permutations: Ordered arrangements of r objects selected from n distinct objects without replacement (r ≤ n)
Stage 1: n possible objects
Stage 2: n-1 remaining possibilities
Stage r : n-r+1 remaining possibilities

7. Counting Rules for Probability (III)
Combinations: Unordered arrangements of r objects selected from n distinct objects without replacement (r ≤ n)
The number of distinct orderings among a set of r objects is r !
# of combinations where order does not matter = number of permutations divided by r !

8. Counting Rules for Probability (IV)
Partitions: Unordered arrangements of n objects partitioned into k groups of size n1,...nk (where n nk = n)
Stage 1: # of Combinations of n1 elements from n objects
Stage 2: # of Combinations of n2 elements from n-n1 objects
Stage k: # of Combinations of nk elements from nk objects

9. Counting Rules for Probability (V)
Runs of binary outcomes (String of m+n trials)
Observe n Successes (S)
Observe m Failures (F)
k = minimum # of “runs” of S or F in the ordered outcomes of the m+n trials

10. Runs Examples – 2006/7 UF NCAA Champs

11. Conditional Probability and Independence
In many situations one event in a multi-stage experiment occurs temporally before a second event
Suppose event A can occur prior to event B
We can consider the probability that B occurs given A has occurred (or vice versa)
For B to occur given A has occurred:
At first stage A has to have occurred
At second stage, B has to occur (along with A)
Assuming P(A), P(B) > 0, we can obtain “Probability of B Given A” and “Probability of A Given B” as follow:
If P(B|A) = P(B) and P(A|B) = P(A), A and B are said to be INDEPENDENT

12. Rules of Probability

13. Bayes’ Rule - Updating Probabilities
Let A1,…,Ak be a set of events that partition a sample space such that (mutually exclusive and exhaustive):
each set has known P(Ai) > 0 (each event can occur)
for any 2 sets Ai and Aj, P(Ai and Aj) = 0 (events are disjoint)
P(A1) + … + P(Ak) = 1 (each outcome belongs to one of events)
If C is an event such that
0 < P(C) < 1 (C can occur, but will not necessarily occur)
We know the probability C will occur given each event Ai: P(C|Ai)
Then we can compute probability of Ai given C occurred:

14. Example - OJ Simpson Trial
Given Information on Blood Test (T+/T-)
Sensitivity: P(T+|Guilty)=1
Specificity: P(T-|Innocent)=.9957  P(T+|I)=.0043
Suppose you have a prior belief of guilt: P(G)=p*
What is “posterior” probability of guilt after seeing evidence that blood matches: P(G|T+)?
Source: B.Forst (1996). “Evidence, Probabilities and Legal Standards for Determination of Guilt: Beyond the OJ Trial”, in Representing OJ: Murder, Criminal Justice, and the Mass Culture, ed. G. Barak pp Harrow and Heston, Guilderland, NY

15. OJ Simpson Posterior (to Positive Test) Probabilities

16. Northern Army at Battle of Gettysburg
Regiments: partition of soldiers (A1,…,A9). Casualty: event C
P(Ai) = (size of regiment) / (total soldiers) = (Column 3)/95369
P(C|Ai) = (# casualties) / (regiment size) = (Col 4)/(Col 3)
P(C|Ai) P(Ai) = P(Ai and C) = (Col 5)*(Col 6)
P(C)=sum(Col 7)
P(Ai|C) = P(Ai and C) / P(C) = (Col 7)/.2416

17. CRAPS Player rolls 2 Dice (“Come out roll”): After first roll:
2,3,12 - Lose (Miss Out)
7,11 - Win (Pass)
4,5,6.8,9,10 - Makes point. Roll until point (Win) or 7 (Lose)
Probability Distribution for first (any) roll:
After first roll:
P(Win|2) = P(Win|3) = P(Win|12) = 0
P(Win|7) = P(Win|11) = 1
What about other conditional probabilities if make point?

18. CRAPS Suppose you make a point: (4,5,6,8,9,10)
You win if your point occurs before 7, lose otherwise and stop
Let P mean you make point on a roll
Let C mean you continue rolling (neither point nor 7)
You win for any of the mutually exclusive events:
P, CP, CCP, …, CC…CP,…
If your point is 4 or 10, P(P)=3/36, P(C)=27/36
By independence, and multiplicative, and additive rules:

19. CRAPS Similar Patterns arise for points 5,6,8, and 9:
For 5 and 9: P(P) = 4/36 P(C) = 26/36
For 6 and 8: P(P) = 5/36 P(C) = 25/36
Finally, we can obtain player’s probability of winning:

20. CRAPS - P(Winning)
Note in the previous slides we derived P(Win|Roll), we multiply those by P(Win) to obtain P(Roll&Win) and sum those for P(Win). The last column gives the probability of each come out roll given we won.

21. Odds, Odds Ratios, Relative Risk
Odds: Probability an event occurs divided by probability it does not occur odds(A) = P(A)/P(Ā)
Many gambling establishments and lotteries post odds against an event occurring
Odds Ratio (OR): Odds of A occurring for one group, divided by odds of A for second group
Relative Risk (RR): Probability of A occurring for one group, divided by probability of A for second group

22. Example – John Snow Cholera Data
2 Water Providers: Southwark & Vauxhall (S&V) and Lambeth (L)
Provider\Status
Cholera Death
No Cholera Death
Total
S&V
3706
263919
267625
Lambeth
411
171117
171528
4117
435036
439153