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PROBABILITY
RULES
CHANCE IS ALL AROUND YOU….
 YOU AND YOUR FRIEND PLAY ROCK-PAPER-SCISSORS TO DETERMINE
WHO GETS THE LAST SLICE OF PIZZA….
 A COIN TOSS DECIDES WHICH TEAM GETS TO RECEIVE THE BALL
FIRST IN THE BEARS GAME….
 ADULTS REGULARLY PLAY THE LOTTO IN HOPES OF
WINNING A BIG JACKPOT….
 OTHERS HEAD TO CASINOS AND RACETRACKS HOPING THAT SOME
COMBINATION OF LUCK AND SKILL WILL PAY OFF…
 PEOPLE YOUNG AND OLD PLAY GAMES OF CHANCE INVOLVING
CARDS OR DICE OR SPINNERS
 THE TRAITS THAT CHILDREN INHERIT – GENDER, EYE COLOR, HAIR
COLOR, DIMPLES, CLEFTS – ARE DETERMINED BY THE CHANCE
INVOLVED IN WHICH GENES GET PASSED ALONG BY THEIR PARENTS
A ROLL OF A DICE…
A SIMPLE RANDOM SAMPLE…
AND EVEN THE INHERITANCE OF YOUR GOOD
LOOKS REPRESENT CHANCE BEHAVIOR THAT
WE CAN UNDERSTAND AND WORK WITH…..
WE CAN
ROLL THE DICE
AGAIN
AND AGAIN
AND AGAIN…..
THE OUTCOMES ARE
GOVERNED BY CHANCE…
BUT IN MANY REPITITIONS
A PATTERN EMERGES….
WE USE
MATHEMATICS
TO UNDERSTAND
THE REGULAR PATTERNS
OF CHANCE BEHAVIOR
WHEN WE CAN REPEAT
THE SAME CHANCE PHENOMENON
AGAIN
AND
AGAIN
THE MATHEMATICS OF
CHANCE IS CALLED
PROBABILITY
AND WE WILL SPEND THESE LAST
5 WEEKS OF STATS STUDYING
PROBABILITY
WELCOME
TO
PROBABILITY!
IN
FOOTBALL
A COIN TOSS
HELPS DETERMINE WHICH TEAM
GETS THE BALL FIRST.
WHY DO THE RULES OF FOOTBALL REQUIRE A COIN TOSS?
TOSSING A COIN
AVOIDS FAVORTISM
FAVORTISM
IS
UNDESIRABLE
THAT’S WHY STATISTICIANS
RECOMMEND
RANDOMIZED EXPERIMENTS
THEY AVOID FAVORTISM
BY LETTING
CHANCE
DECIDE WHO GETS
CHOSEN…
A BIG FACT EMERGES WHEN WE
WATCH COIN TOSSES OR THE
RESULTS OF RANDOM EXPERIMENTS
CLOSELY
CHANCE BEHAVIOR
IS UNPREDICATABLE
IN THE SHORT RUN, BUT HAS A
REGULAR PATTERN IN THE
LONG RUN.
WHEN YOU FLIP A COIN IT IS
EQUALLY LIKELY TO LAND
“HEADS” OR “TAILS”
DO
THUMBTACKS
BEHAVE IN THE
SAME
WAY?
IN THIS ACTIVITY
YOU WILL
TOSS A THUMBTACK
SEVERAL TIMES AND
OBSERVE
WHETHER IT COMES TO
REST
POINT UP (U)
OR
POINT DOWN(D)
THE QUESTION YOU ARE TRYING TO
ANSWER IS…
“WHAT IS THE PROBABILITY THAT
THE TOSSED THUMBTACK WILL
LAND POINT DOWN?”
MAKE A GUESS...
WHAT PERCENTAGE OF THE
TIME DO YOU THINK
A THUMBTACK
WILL LAND
POINT DOWN?
WHEN TOSSED ONTO A FLAT SURFACE A COMMON
THUMBTACK CAN “POINT UP” OR “POINT DOWN”.
WE ARE CURIOUS, IF LIKE A FAIR COIN,
THESE TWO OUTCOMES ARE
EQUALLY LIKELY
AND IF NOT,
WHAT IS THE PROBABILITY OF EACH OUTCOME?
ACCORDING TO THE
LAW OF LARGE NUMBERS
THE LONG-RUN RELATIVE FREQUENCY
OF REPEATED
INDEPENDENT EVENTS
GETS
CLOSER AND CLOSER
TO THE
TRUE RELATIVE FREQUENCY
EVEN THOUGH THE
LAW OF LARGE NUMBERS
SEEMS NATURAL
IT IS OFTEN MISUNDERSTOOD BECAUSE THE IDEA OF
LONG RUN
IS HARD TO GRASP!
MANY PEOPLE BELIEVE THAT AN OUTCOME OF A RANDOM
EVENT THAT HASN’T OCCURRED IN MANY TRIALS IS
“DUE”
TO OCCUR….
TURN TO YOUR PARTNER AND GIVE THEM AN EXAMPLE OF
THE NONEXISTENT LAW OF AVERAGES NOW!
A GOOD HITTER IN BASEBALL PLAYER STRIKES OUT 6 TIMES IN A
ROW SO HE IS “DUE” FOR A HIT NEXT TIME UP, RIGHT??
A GIRL WILL BE SURE TO BE BORN NEXT SINCE THERE
ARE 5 BOYS IN A FAMILY, RIGHT??
THE BALL WILL LAND ON RED ON THE ROULETTE WHEEL SINCE
IT HAS LANDED ON BLACK THE LAST 11 TIMES, RIGHT??
YOU JUST FLIPPED 5 HEADS IN A ROW…THE COIN
“OWES” YOU A TAIL, RIGHT??
RIGHT????????????
WRONG
THE
LAW OF LARGE NUMBERS
SAYS
NOTHING
ABOUT
SHORT RUN BEHAVIOR.
RELATIVE FREQUENCIES
EVEN OUT ONLY IN
THE LONG RUN!
( AND, ACCORDING TO THE LLN, THE LONG RUN IS
REALLY LONG – INFINITELY LONG IN FACT!)
THIS SO CALLED
LAW OF AVERAGES
DOESN’T EXIST AT ALL.
PLEASE OPEN YOUR
PACKET UP TO PAGE 1
A RANDOM PHENOMENON IS CONSIDERED TO BE ANY ACT
WHICH______________________________.
FOR EXAMPLE, ROLLING A DIE IS CONSIDERED A RANDOM
PHENOMENON.
TURN TO YOUR PARTNER AND
GIVE ANOTHER EXAMPLE
A RANDOM PHENOMENON IS CONSIDERED TO BE ANY ACT
WHICH CAN HAVE A RANDOM RESULT.
FOR EXAMPLE, ROLLING A DIE IS CONSIDERED A RANDOM
PHENOMENON.
TURN TO YOUR PARTNER AND
GIVE ANOTHER EXAMPLE
A TRIAL IS A _________ ATTEMPT OF A RANDOM PHENOMENON.
ROLLING A DIE ONCE FOR EXAMPLE IS
ONE TRIAL OF A RANDOM PHENOMENON.
A TRIAL IS A SINGLE ATTEMPT OF A RANDOM PHENOMENON.
ROLLING A DIE ONCE FOR EXAMPLE IS
ONE TRIAL OF A RANDOM PHENOMENON.
AN OUTCOME IS ______________________ FROM ANY TRIAL OF A
RANDOM PHENOMENON.
FOR EXAMPLE, IF THE RANDOM PHENOMENON IS ROLLING A
DIE, THEN ROLLING IT ONCE IS A _______AND THERE ARE ______
DIFFERENT OUTCOMES FOR THIS RANDOM PHENOMENON.
ANOTHER EXAMPLE IS ROLLING A DIE AND FLIPPING A COIN.
THIS COULD RESULT IN _________DIFFERENT OUTCOMES.
AN OUTCOME IS A POSSIBLE RESULT FROM ANY TRIAL OF A
RANDOM PHENOMENON.
FOR EXAMPLE, IF THE RANDOM PHENOMENON IS ROLLING A
DIE, THEN ROLLING IT ONCE IS A TRIAL AND THERE ARE SIX
DIFFERENT OUTCOMES FOR THIS RANDOM PHENOMENON.
ANOTHER EXAMPLE IS ROLLING A DIE AND FLIPPING A COIN.
THIS COULD RESULT IN TWELVE DIFFERENT OUTCOMES.
THE SAMPLE SPACE IS ______________________________ OF RANDOM
PHENOMENON. THE SAMPLE SPACE FOR ROLLING A DIE IS THE
SIX OUTCOMES
.
____________________________________________
THE SAMPLE SPACE IS THE SET OF ALL POSSIBLE OUTCOMES OF
RANDOM PHENOMENON. THE SAMPLE SPACE FOR ROLLING A DIE
IS THE SIX OUTCOMES
1, 2, 3, 4, 5, 6.
AN EVENT IS__________________________________.
TECHNICALLY, AN EVENT IS ANY SUBSET OF THE SAMPLE
SPACE. FOR OUR PURPOSES, WE WILL USUALLY TREAT
EVENT AND OUTCOME AS THE SAME.
AN EVENT IS ANY OUTCOME OR SET OF OUTCOMES.
TECHNICALLY, AN EVENT IS ANY SUBSET OF THE SAMPLE
SPACE. FOR OUR PURPOSES, WE WILL USUALLY TREAT
EVENT AND OUTCOME AS THE SAME.
THE EQUALLY LIKELY CONDITION SAYS
________________________________________
_________________________________________.
GIVE SOME EXAMPLES TO YOUR PARTNER…
THE EQUALLY LIKELY CONDITION SAYS
THE OUTCOMES BEING COUNTED ARE ALL
EQUALLY LIKELY TO OCCUR.
GIVE SOME EXAMPLES TO YOUR PARTNER…
ACCORDING TO THE
LAW OF LARGE NUMBERS
THE LONG-RUN RELATIVE FREQUENCY
OF REPEATED
INDEPENDENT EVENTS
GETS
CLOSER AND CLOSER
TO THE
TRUE RELATIVE FREQUENCY
MANY PEOPLE BELIEVE THAT AN OUTCOME OF A RANDOM EVENT THAT
HASN’T OCCURRED IN MANY TRIALS IS
“DUE”
TO OCCUR….
THIS IS REFERRED TO AS
THE LAW OF AVERAGES
IT IS NOT TRUE!!!
THE FUNDAMENTAL COUNTING PRINCIPLE
PART 1
OR
IF EVENT A HAS “M” OUTCOMES AND EVENT B HAS “N” DIFFERENT
OUTCOMES THEN THE NUMBER OF OUTCOMES
IN EVENT
A OR B
IS
M+N
YOU HAVE 6 X-BOX GAMES AND 10 WII GAMES.
IN HOW MANY WAYS CAN YOU CHOOSE
AN X-BOX GAME
OR
A WII GAME
TO PLAY AFTER SCHOOL?
ANSWER: 16 WAYS
THE FUNDAMENTAL COUNTING PRINCIPLE
PART 2
AND
IF EVENT A HAS “ M” OUTCOMES AND EVENT B HAS “ N” DIFFERENT
OUTCOMES THEN THE NUMBER OF OUTCOMES
IN EVENT
A AND B
IS
M•N
YOU HAVE 6 X-BOX GAMES AND 10 WII GAMES.
IN HOW MANY WAYS CAN YOU CHOOSE
AN X-BOX GAME
AND
A WII GAME
TO PLAY AFTER SCHOOL?
ANSWER: 60 WAYS
DEFINITION OF PROBABILITY
P(A) = # OF OUTCOMES IN A
# OF POSSIBLE EQUALLY LIKELY OUTCOMES
WHAT’S THE PROBABILITY
OF DRAWING A FACE CARD?
DEFINITION OF PROBABILITY
P(A) = # OF OUTCOMES IN A
# OF POSSIBLE EQUALLY LIKELY OUTCOMES
EXAMPLE: PROBABILITY OF DRAWING A FACE CARD
P(FACE CARD) = # FACE CARDS
# CARDS
= 12/52
= 3/13
GIVEN 2 SIX-SIDED FAIR DICE, WHAT’S
THE PROBABILITY OF ROLLING THE SUM
OF 7?
FIRST, MAKE A SAMPLE SPACE
LISTING ALL THE POSSIBLE OUTCOMES
THESE ARE THE TOTALS:
2,3,4,5,6,7,8,9,10,11,12
BUT
THEY ARE NOT ALL
EQUALLY LIKELY…
WHY NOT???
MAKE A TABLE OF OUTCOMES
SUPPOSE A FAMILY HAS TWO CHILDREN.
LIST THEY OUTCOMES IN THE SAMPLE
SPACE
BB, BG, GB, GG
BB,BG,GB,GG
A) WHAT ARE WE ASSUMING IN THINKING THESE FOUR OUTCOMES ARE
EQUALLY LIKELY?
B) WHAT IS THE PROBABILITY A 2-CHILD FAMILY HAS TWO GIRLS?
C) WHAT IS THE PROBABILITY THERE IS AT LEAST ONE GIRL?
D) WHAT IS THE PROBABILITY BOTH CHILDREN ARE THE SAME SEX?
BB,BG,GB,GG
A) WHAT ARE WE ASSUMING IN THINKING THESE FOUR OUTCOMES ARE
EQUALLY LIKELY? THE CHANCE OF HAVING A BOY/GIRL IS 50 – 50 AND
THE SEXES OF BABIES BORN IN THE SAME FAMILY ARE INDEPENDENT
B) WHAT IS THE PROBABILITY A 2-CHILD FAMILY HAS TWO GIRLS? 1/4
C) WHAT IS THE PROBABILITY THERE IS AT LEAST ONE GIRL?
3/4
D) WHAT IS THE PROBABILITY BOTH CHILDREN ARE THE SAME SEX?
2/4 OR 1/2
TONIGHT’S HOMEWORK
READ PAGES 284 – 290
AND
DO PAGE 300
#1,3,4,6,8,10,17,23,