WARMUP

•WRITE DOWN 5 WAYS IN WHICH YOU SEE/USE PROBABILITY IN EVERY DAY LIFE.

CHAPTER 3: PROBABILITY

3.1 BASIC CONCEPTS OF PROBABILITY AND COUNTING

VOCABULARY • PROBABILITY EXPERIMENT: AN ACTION, OR TRIAL, THROUGH

WHICH SPECIFIC RESULTS (COUNTS, MEASURES, OR RESPONSES) ARE OBTAINED.

• OUTCOME: THE RESULT OF A SINGLE TRIAL OF A PROBABILITY EXPERIMENT

• SAMPLE SPACE: THE SET OF ALL POSSIBLE OUTCOMES IN A SAMPLE SPACE. EVENTS ARE SUBSETS OF THE SAMPLE SPACE.

DETERMINE THE NUMBER OF OUTCMES AND IDENTIFY THE SAMPLE SPACE

• DRAWING A DIAMOND OUT OF A DECK OF CARDS

• ROLLING A 2 ON A 6-SIDED DIE.

• DRAWING A KING OUT OF A DECK OF CARDS

• FLIPPING HEADS ON A COIN

• DRAWING A BLACK 4 FROM A DECK OF CARDS.

SOMETIMES WE CAN DRAW OUT THE NUMBER OF RESPONSES USING A DIAGRAM

• SUPPOSE WE PROPOSED THE QUESTION:

• DOES YOUR FAVORITE TEAM’S WIN OR LOSS AFFECT YOUR MOOD?

• A. YES B. NO. C. NOT SURE

• LET’S SAY WE WANTED TO DIVIDE THE RESPONSES UP BY GENDER AS WELL. WHAT WOULD OUR SAMPLE SPACE LOOK LIKE? HOW BIG WOULD IT BE?

THE FUNDAMENTAL COUNTING PRINCIPLE

• IF EVENT 1 CAN OCCUR M WAYS AND EVENT 2 CAN OCCUR N WAYS, THEN BOTH EVENTS CAN OCCUR A NUMBER OF M X N WAYS. WE USE THIS RULE FOR EVENTS OCCURRING IN SEQUENCE.

EXAMPLE…

• SUPPOSE YOU TOOK YOUR DATE OUT FOR A NICE STEAK DINNER. YOU HAVE THE FOLLOWING OPTIONS FOR YOUR CHOICE OF STEAK AND A SIDE.

• CUT OF STEAK: RIBEYE, T-BONE, NEW YORK STRIP, FILET MIGNON, PORTERHOUSE

• TEMPERATURE: RARE, MEDIUM RARE, MEDIUM WELL, WELL-DONE

• SIDE: LOADED BAKE POTATOES, SWEET POTATOES, RICE PILAF, MACORONI AND CHEESE, SALAD.

• USE THE FUNDAMENTAL COUNTING PRINCIPLE TO FIND OUT HOME MANY COMBINATIONS YOU CAN MAKE.

YOUR BANK ACCOUNT HAS A 4-DIGIT PIN NUMBER…

• HOW MANY POSSIBLE CODES ARE THERE IF YOU CAN REPEAT EVERY NUMBER?

PIN NUMBER CONTINUED…

• WHAT IF NUMBERS COULD NOT BE REPEATED AND ONLY USED ONCE?

CLASSICAL PROBABILITY

• CLASSICAL PROBABILITY IS USED WHEN EACH OUTCOME IN A SAMPLE IS EQUALLY LIKELY TO OCCUR.

• P(E)=

• WHAT IS THE PROBABILITY OF ROLLING A THREE ON A SIX SIDED DIE?

SUPPOSE YOU ROLLED A SIX-SIDED DIE

• WHAT IS THE PROBABILITY YOU ROLL AN EVEN NUMBER?

• A NUMBER GREATER THAN 4?

WHAT IS THE PROBABILITY OF LANDING ON A RED SPACE? (THERE ARE 24 SPACES)

EMPIRICAL PROBABILITY

• EMPIRICAL PROBABILITY: BASED ON OBSERVATIONS OBTAINED FROM PROBABILITY EXPERIMENTS.

• THE FORMULA… P(E)=

EXAMPLE…• SUPPOSE THE FOLLOWING FREQUENCY DISTRIBUTION REPRESENT GRADES

ON A TEST…

• FIND THE PROBABILITY THAT YOU MADE A C ON THE TEST…

• FIND THE PROBABILITY THAT YOU MADE AN A OR B ON THE TEST…GRADE FREQUENCY

A 5

B 7

C 16

D 4

F 3

LAST PART OF 3.1• LAW OF LARGE NUMBERS: REPETITION OF AN EXPERIMENT SO THAT THE

EMPIRICAL PROBABILITY OF AN EVENT APPROACHES THE ACTUAL PROBABILITY OF THE EVENT.

• COMPLEMENT OF EVENT E: SET OF ALL OUTCOMES IN A SAMPLE SPACE THAT ARE NOT INCLUDED IN EVENT E. DENOTED AS E’.

• LOOKING BACK AT THE TEST SCORES TABLE. WHAT IS THE PROBABILITY THAT YOU DID NOT MAKE A C ON THE TEST?

GRADE FREQUENCY

A 5

B 7

C 16

D 4

F 3

SECTION 3.2CONDITIONAL PROBABILITY AND THE MULTIPLICATION RULE

CONDITIONAL PROBABILITY

• PROBABILITY OF AN EVENT OCCURING, GIVEN THAT ANOTHER HAS ALREADY OCCURRED.

• READ AS “PROBABILITY OF B, GIVEN A.” AND DENOTED BY P(B|A)

• YOU ARE DEALT TWO CARDS FROM A 52-CARD DECK, ONE FACE UP, AND ONE FACE DOWN. IF THE CARD SHOWING IS AN ACE, WHAT IS THE PROBABILITY THAT THE CARD FACE DOWN IS AN ACE AS WELL?

EXAMPLE 2…• THE RESULTS OF A MIDDLE SCHOOL SURVEY ON FAVORITE WINTER SPORTS

ARE AS FOLLOWS:

• WHAT IS THE PROBABILITY OF SELECTING A 6TH GRADER?

• WHAT IS THE PROBABILITY OF SELECTING A 7TH GRADER THAT PREFERS SNOWBOARDING?

• SUPPOSE THE STUDENT SELECTED LIKES SKIING, WHAT IS THE PROBABILITY THAT IT IS AN 8TH GRADER?

TYPES OF EVENTS AND THE MULTIPLICATION RULE

• TWO EVENTS ARE INDEPENDENT IF THE OCCURRENCE OF ONE EVENT DOES NOT AFFECT THE PROBABILITY OF THE SECOND EVENT. OTHERWISE THEY ARE SAID TO BE DEPENDENT EVENTS.

• INDEPENDENT EVENTS ARE DENOTED AS P(B|A)=P(B) OR IF P(A|B)=P(A)

• DETERMINE IF THE FOLLOWING ARE DEPENDENT OR INDEPENDENT EVENTS…

• SELECTING A KING FROM A DECK OF CARDS, REPLACING IT, AND THEN SELECTING AN ODD NUMBER.

• TOSSING A COIN AND THEN ROLLING A 6-SIDED DIE

• NOT DOING YOUR HOMEWORK AND TRYING TO PASS THE CLASS.

• WINNING THE VOLLEYBALL GAME AND THE WEATHER BEING SUNNY.

MULTIPLICATION RULE

•PROBABILITY THAT TWO EVENTS WILL OCCUR IN SEQUENCE IS:

•P(A AND B)= P(A) X P(B|A). IF THE EVENTS ARE INDEPENDENT THE FORMULA CAN SIMPLY BE WRITTEN AS P(A) X P(B).

EXAMPLES…• PROBABILITY OF GETTING A 10 DEALT TO YOU AND THEN

GETTING AN ACE DEALT TO YOU.

• P(10)=…. P(A)=…

• THE PROBABILITY THAT A WEIGHT LOSS PILL IS SUCCESSFUL IS 71%.

• WHAT IS THE PROBABILITY THAT IT IS SUCCESSFUL ON 4 PEOPLE?

• WHAT IS THE PROBABILITY THAT IT IS SUCCESSFUL ON 0 OF THE 4 PEOPLE?

• WHAT IS THE PROBABILITY THAT IT IS SUCCESSFUL ON AT LEAST 1 OF THE 4 PEOPLE?

THE ADDITION RULESECTION 3-3

MUTUALLY EXCLUSIVE EVENTS

• EVENTS THAT OCCUR IN SEQUENCE HAVE A PROBABILITY DENOTED BY P(A AND B).

• MUTUALLY EXCLUSIVE EVENTS ARE DENOTED BY P(A OR B).

• TWO EVENTS ARE MUTUALLY EXCLUSIVE IF EVENT A AND EVENT B CANNOT OCCUR AT THE SAME TIME

• WE USE MUTUALLY EXCLUSIVE EVENTS TO FIND THE PROBABILITY THAT AT LEAST ONE EVENT WILL OCCUR.

EXAMPLES…• DECIDE IF THE FOLLOWING EVENTS ARE MUTUALLY

EXCLUSIVE. CAN THEY OCCUR AT THE SAME TIME? DO THEY HAVE ANY OUTCOMES IN COMMON?

• EVENT A: SELECTING A JACK FROM A STANDARD DECK OF CARDS

• EVENT B: SELECTING A FACE CARD FROM A STANDARD DECK OF CARDS

• EVENT A: RANDOMLY SELECTING A VEHICLE THAT IS A FORD

• EVENT B: RANDOMLY SELECTING A VEHICLE THAT IS A TOYOTA

THE ADDITION RULE

• THE PROBABILITY THAT EVENTS A OR BB WILL OCCUR P(A OR B) IS GEVEN BY

• P(A OR B)= P(A)+P(B)-P(A AND B)

• IF THEY ARE MUTUALLLY EXCLUSIVE, THEN OUR FORMULA IS SIMPLY P(A OR B)= P(A)+P(B)

P(A OR B)= P(A)+P(B)-P(A AND B)

• YOU WIN $10 IF YOU ARE DEALT A QUEEN OR A HEART FROM A STANDARD DECK OF CARDS. FIND THE PROBABILITY THAT YOU WIN THE GAME.

Blood Types

O A B AB TOTAL

Positive 156 139 37 12 344

Negative 28 25 8 4 65

Total 184 164 45 16 409

TYPE OF PROBABILITY FORMULA/SYMBOLS

CLASSICAL PROBABILITY

EMPIRICAL PROBABILITY

RANGE OF PROBABILITIES RULE

COMPLMENTARY EVENTS

MULTIPLICATION RULE

ADDITION RULE