PROBABILITY AND PROBABILITY RULES CHAPTER 4 SECTIONS 4.1 – 4.2 PROBABILITY AND PROBABILITY RULES

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

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

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

EXAMPLE 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.

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}

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

Probability of an Event, E LET E BE ANY EVENT AND S THE SAMPLE SPACE. THE PROBABILITY OF E, DENOTED P(E) IS COMPUTED AS

Probability of an Event The probability of an event A, denoted by P(A), is obtained by adding the probabilities of the individual outcomes in the event. When all the possible outcomes are equally likely,

Probability Model ILLUSTRATION: Tossing of a coin A probability model is a mathematical description of long-run regularity consisting of a sample space S and a way of assigning probabilities to events. ILLUSTRATION: Tossing of a coin

Probability Rules We can use graphs to represent probabilities and probability rules. Venn diagram: Area = Probability S A

Probability Rules Rule 1: A probability is a number between 0 and 1. For any event A, 0 ≤ P(A) ≤ 1. A probability can be interpreted as the proportion of times that a certain event can be expected to occur. If the probability of an event is more than 1, then it will occur more than 100% of the time (Impossible!).

Probability Rules Rule 2: All possible outcomes together must have probability 1. P(S) = 1. Because some outcome must occur on every trial, the sum of the probabilities for all possible outcomes must be exactly one. If the sum of all of the probabilities is less than one or greater than one, then the resulting probability model will be incoherent.

Illustration of Rule 2

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.

Probability Rules Rule 3 (Complement Rule): The probability that an event does not occur is 1 minus the probability that the event does occur. The set of outcomes that are not in the event A is called the complement of A, and is denoted by AC. P(AC) = 1 – P(A). Example: what is the probability of getting at least 1 head in the experiment of tossing a fair coin 3 times? HHH HHT HTH HTT THH THT TTH TTT

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:

Probability Rules Complement rule: P(AC) = 1 – P(A). A AC

Complement of an Event, A

Probability Rules Rule 4 (Multiplication Rule): Two events are said to be independent if the occurrence of one event does not influence the occurrence of the other. For two independent events A and B, the probability that both A and B occur is the product of the probabilities of the two events. P(A and B) = P(A) X P(B).

Examples From Handout

Combining 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.

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.

Intersection and Union of Events

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),

Probability Rules Rule 5 (Addition Rule): If two events have no outcomes in common, they are said to be disjoint (or mutually exclusive). The probability that one or the other of two disjoint events occurs is the sum of their individual probabilities. P(A or B) = P(A) + P(B), provided that A and B are disjoint.

Addition Rule Addition rule: P(A or B) = P(A) + P(B), A and B are disjoint. S A B

Example From Handout

Probability Rule 6 – The General Addition Rule For two events A and B, the probability that A or B occurs equals the probability that A occurs plus the probability that B occurs minus the probability that A and B both occur. P(A or B) = P(A) + P(B) – P(A and B).

Probability of the Union of Events

Examples From Handout

Solution

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

Extra-Credit – Question 1 In a large Statistics lecture, the professor reports that 52% of the students enrolled have never taken a Calculus course, 34% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. (i) What is the probability that the first group member you meet has studied some Calculus? (ii) What is the probability that the first group member you meet has studied no more than one semester of Calculus? (iii) What is the probability that both of your two group members have studied exactly one semester of Calculus? (iv) What is the probability that at least one of your group members has had more than one semester of Calculus?

Extra Credit – Questions 2 and 3 A PAIR OF FAIR DICE IS TOSSED. FIND THE PROBABILITY THAT THE MAXIMUM OF THE TWO NUMBERS IS GREATER THAN 4. Question 3 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.