1. PROBABILITY, PROBABILITY RULES, AND CONDITIONAL PROBABILITY
CHAPTERS 5
PROBABILITY, PROBABILITY RULES, AND CONDITIONAL PROBABILITY

2. PROBABILITY MODELS: FINITELY MANY OUTCOMES
DEFINITION:
PROBABILITY IS THE STUDY OF RANDOM OR NONDETERMINISTIC EXPERIMENTS. IT MEASURES THE NATURE OF UNCERTAINTY.

3. PROBABILISTIC TERMINOLOGIES
RANDOM EXPERIMENT
AN EXPERIMENT IN WHICH ALL OUTCOMES (RESULTS) ARE KNOWN BUT SPECIFIC OBSERVATIONS CANNOT BE KNOWN IN ADVANCE.
EXAMPLES:
TOSS A COIN
ROLL A DIE

4. SAMPLE SPACE
THE SET OF ALL POSSIBLE OUTCOMES OF A RANDOM EXPERIMENT IS CALLED THE SAMPLE SPACE.
NOTATION: S
EXAMPLES
FLIP A COIN THREE TIMES
S =

5. EXAMPLE 2. RANDOM VARIABLE
AN EXPERIMENT CONSISTS OF FLIPPING A COIN AND THEN FLIPPING IT A SECOND TIME IF A HEAD OCCURS. OTHERWISE, ROLL A DIE.
RANDOM VARIABLE
THE OUTCOME OF AN EXPERIMENT IS CALLED A RANDOM VARIABLE. IT CAN ALSO BE DEFINED AS A QUANTITY THAT CAN TAKE ON DIFFERENT VALUES.

6. S = {HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}
EXAMPLE
FLIP A COIN THREE TIMES. IF X DENOTES THE OUTCOMES OF THE THREE FLIPS, THEN X IS A RANDOM VARIABLE AND THE SAMPLE SPACE IS
S = {HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}
IF Y DENOTES THE NUMBER OF HEADS IN THREE FLIPS, THEN Y IS A RANDOM VARIABLE. Y = {0, 1, 2, 3}

7. PROBABILITY DISTRIBUTION
LET X BE A RANDOM VARIABLE WITH ASSOCIATED SAMPLE SPACE S. A PROBABILITY DISTRIBUTION (p. d.) FOR X IS A FUNCTION P WHOSE DOMAIN IS S, WHICH SATISFIES THE FOLLOWING TWO CONDITIONS:
0 ≤ P (w) ≤ 1 FOR EVERY w IN S.
P (S) = 1, I.E. THE SUM OF P(S) IS
ONE.

8. REMARKS
IF P (w) IS CLOSE TO ZERO, THEN THE OUTCOME w IS UNLIKELY TO OCCUR.
IF P (w) IS CLOSE TO 1, THE OUTCOME w IS VERY LIKELY TO OCCUR.
A PROBABILITY DISTRIBUTION MUST ASSIGN A PROBABILITY BETWEEN 0 AND 1 TO EACH OUTCOME.
THE SUM OF THE PROBABILITY OF ALL OUTCOMES MUST BE EXACTLY 1.

9. EXAMPLES
A COIN IS WEIGHTED SO THAT HEADS IS TWICE AS LIKELY TO APPEAR AS TAILS. FIND P(T) AND P(H).
2. THREE STUDENTS A, B AND C ARE IN A SWIMMING RACE. A AND B HAVE THE SAME PROBABILITY OF WINNING AND EACH IS TWICE AS LIKELY TO WIN AS C. FIND THE PROBABILITY THAT B OR C WINS.

10. EVENTS
AN EVENT IS A SUBSET OF A SAMPLE SPACE, THAT IS, A COLLECTION OF OUTCOMES FROM THE SAMPLE SPACE.
EVENTS ARE DENOTED BY UPPER CASE LETTERS, FOR EXAMPLE, A, B, C, D.
LET E BE AN EVENT. THEN THE PROBABILITY OF E, DENOTED P(E), IS GIVEN BY

11. FOR ANY EVENT E, 0 < P(E) < 1
COMPUTATIONAL FORMULA
LET E BE ANY EVENT AND S THE SAMPLE SPACE. THE PROBABILITY OF E, DENOTED P(E) IS COMPUTED AS

12. EXAMPLES
A PAIR OF FAIR DICE IS TOSSED. FIND THE PROBABILITY THAT THE MAXIMUM OF THE TWO NUMBERS IS GREATER THAN 4.
ONE CARD IS SELECTED AT RANDOM FROM 50 CARDS NUMBERED 1 TO 50. FIND THE PROBABILITY THAT THE NUMBER ON THE CARD IS (I) DIVISIBLE BY 5, (II) PRIME, (III) ENDS IN THE DIGIT 2.

13. NULL EVENT: AN EVENT THAT HAS NO CHANCE OF OCCURING
NULL EVENT: AN EVENT THAT HAS NO CHANCE OF OCCURING. THE PROBABILITY OF A NULL EVENT IS ZERO P( NULL EVENT ) = 0
CERTAIN OR SURE EVENT: AN EVENT THAT IS SURE TO OCCUR. THE PROBABILITY OF A SURE OR CERTAIN EVENT IS ONE.
P(S) = 1

14. COMBINATION OF EVENTS INTERSECTION OF EVENTS
THE INTERSECTION OF TWO EVENTS A AND B, DENOTED
IS THE EVENT CONTAINING ALL ELEMENTS(OUTCOMES) THAT ARE COMMON TO A AND B.

15. PICTURE DEMONSTRATION

16. UNION OF EVENTS THE UNION OF TWO EVENTS A AND B, DENOTED,
IS THE EVENT CONTAINING ALL THE ELEMENTS THAT BELONG TO A OR B OR BOTH.

17. PICTURE DEMONSTRATION

18. COMPLEMENT OF AN EVENT
THE COMPLEMENT OF AN EVENT A WITH RESPECT TO S IS THE SUBSET OF ALL ELEMENTS(OUTCOMES) THAT ARE NOT IN A.
NOTATION:

19. PICTURE DEMONSTRATION

20. MUTUALLY EXCLUSIVE(DISJOINT) EVENTS
TWO EVENTS A AND B ARE MUTUALLY EXCLUSIVE(DISJOINT) IF
THAT IS, A AND B HAVE NO OUTCOMES IN COMMON.
IF A AND B ARE DISJOINT(MUTUALLY EXCLUSIVE),

21. PICTURE DEMONSTRATION

22. ADDITION RULE GENERAL ADDITION RULE
IF A AND B ARE MUTUALLY EXCLUSIVE EVENTS, THEN
GENERAL ADDITION RULE
IF A AND B ARE ANY TWO EVENTS, THEN

23. PICTURE DEMONSTRATION

24. INDEPENDENCE OF EVENTS
TWO EVENTS A AND B ARE SAID TO BE INDEPENDENT IF ANY OF THE FOLLOWING EQUIVALENT CONDITIONS ARE TRUE:

25. CLASSWORK: EXAMPLES FROM PRACTICE EXERCISES SHEET 2

26. CONDITIONAL PROBABILITY AND DECISION TREES
LET A AND B BE ANY TWO EVENTS FROM A SAMPLE SPACE S FOR WHICH P(B) > 0. THE CONDITIONAL PROBABILITY OF A GIVEN B, DENOTED
IS GIVEN BY

27. CLASSWORK: EXAMPLES FROM PRACTICE EXERCISES SHEET 2

28. GENERAL MULTIPLICATION RULE
THE FORMULA FOR CONDITIONAL PROBABILITY CAN BE MANIPULATED ALGEBRAICALLY SO THAT THE JOINT PROBABILITY P(A and B) CAN BE DETERMINED FROM THE CONDITIONAL PROBABILITY OF AN EVENT. USING
AND SOLVING FOR P(A and B), WE OBTAIN THE GENERAL MULTIPLICATION RULE

29. CLASSWORK: EXAMPLES FROM PRACTICE EXERCISES SHEET 2

30. CONDITIONAL PROBABILITY CONT’D
CONDITIONAL PROBABLITY THROUGH BAYE’S FORMULA
SHALL BE SKIPPED FOR THIS CLASS

31. BAYES’ FORMULA FOR TWO EVENTS A AND B
BY THE DEFINITION OF CONDITIONAL PROBABILITY,

32. CLASSWORK: EXAMPLES FROM PRACTICE EXERCISES SHEET 2