Transcript Game_of_Pig_probability_5th_grd.ppt

THE GAME OF PIG
PROBABILITY
AND
EXPECTED VALUE
THE GAME OF PIG
ROLL A SIX
SIDED DIE
IF YOU
ROLL A 1,
THEN
YOUR
TURN IS
OVER
PIG STRATEGIES
WRITE DOWN THE BEST STRATEGY
WAITING FOR A DOUBLE
A DOUBLE MEANS HAVING THE SAME NUMBER
SHOW ON BOTH DICE
GAMBLERS FALLACY
WE WILL CALL THREE
IDENTICAL FLIPS IN A ROW A
TRIPLET
VOCABULARY
• THEORETICAL PROBABILITY
– IS PROBABILITY DETERMINED ON THE BASIS OF A
MODEL
• OBSERVED PROBABILITY
– IS PROBABILITY BASED ON ACTUAL RESULTS OF
PAST EVENTS
WHAT ARE THE CHANCES?
THEORETICAL
OR
OBSERVED
PAULA’S PIZZA
HOW MANY DIFFERENT
POSSIBILITIES?
RUG GAMES
DO WE HAVE
EQUAL REGIONS?
THE COUNTER’S GAME
PLACE 11 MARKERS
ROLL A PAIR OF DICE
ROLLIN’ ROLLIN’
ROLL A PAIR OF
DICE 50 TIMES
THE THEORY OF
TWO-DICE SUMS
DEVELOP A RUG DIAGRAM
TWO-DICE SUMS AND PRODUCTS
DEVELOP ANOTHER RUG DIAGRAM
SPINNER GIVE AND TAKE
MAKE A
SPINNER
AND PLAY THE
GAME
MIA’S CARDS
A STANDARD DECK
OF CARDS
HEARTS WORTH 10
PTS
CLUB, SPADE,
DIAMONDS WORT
PTS
EXPECTED VALUE
• IS THE AVERAGE AMOUNT GAINED (OR LOST)
PER TURN IN THE LONG RUN
A FAIR RUG GAME?
CRYSTAL WINS
$5
TONY WINS $3
ONE-AND-ONE
60% CHANCE
OF MAKING
THE FREE
THROW
A SIXTY-PERCENT SOLUTION
SIMULATION
40 TIMES
THE THEORY OF ONE-AND-ONE
DEVELOP A RUG DIAGRAM AND FIND THE EXPECTED
VALUE
STREAK-SHOOTING SHELLY
EXPECTED VALUE
SPINS AND DRAWS
ARCIBALD
CHARITY
BEATRICE
MARTIAN BASKETBALL
1ST SHOT
80% 2ND
60%
AND
90% 3RD
THE CARRIER’S PAYMENT PLAN
QUANDARY
$5
OR
PICK
TWO
BILLS
AND RECEIVE
$11
OR
$2
THE GAME OF LITTLE PIG
RED CUBE = 1 POINT
BLUE CUBE = 4 POINTS
YELLOW CUBE = 0 POINTS
LITTLE PIG STRATEGIES
• THE TWO POINT STRATEGY
• THE TWO DRAW STRATEGY
LITTLE PIG STRATEGIES
• THE THREE POINT STRATEGY
• THE THREE DRAW STRATEGY
LITTLE PIG STRATEGIES
• THE FOUR POINT STRATEGY
• THE FOUR DRAW STRATEGY
LITTLE PIG STRATEGIES
• THE FIVE POINT STRATEGY
• THE FIVE DRAW STRATEGY
LITTLE PIG STRATEGIES
• THE SIX POINT STRATEGY
• THE SIX DRAW STRATEGY
SHOULD I GO ON?
WHICH STRATEGY IS
BETTER?
POINT OR DRAW??????
THE BEST LITTLE PIG
FOR WHAT SCORES DOES IT
PAY TO DRAW AGAIN?
FOR WHAT SCORES
DOES IT PAY TO
STOP?
BIG PIG MEETS LITTLE PIG
AFTER HOW
MANY POINTS
SHOULD WE STOP
ROLLING THE
DIE?
SOLVING THE UNIT PROBLEM
SUMMARIZE WHAT
YOU HAVE LEARNED
ABOUT THE BEST
POSSIBLE STRATEGY
FOR PIG.
WAITING FOR A DOUBLE
• MAKE A PREDICTION
• DO THE EXPERIMENT 10 TIMES
– WHAT WAS THE LARGEST # OF ROLLS TO GET A
DOUBLE?
– WHAT WAS THE AVG OF THE TEN EXPERIMENTS?
WAITING FOR A DOUBLE
• HOW CLOSE IS YOUR PREDICTED AVERAGE TO
THE YOUR EXPERIMENTS AVERAGE?
WAITING FOR A DOUBLE
• MAKE A FREQUENCY BAR GRAPH OF THE CLASSES
DATA
THE GAMBLER’S FALLACY
• WHAT DO YOU THINK ABOUT THE STRATEGY
THE PEOPLE USED IN 1913 AT THE CASINO IN
MONTE CARLO?
THE GAMBLER’S FALLACY
• FLIP A COIN 25 TIMES AND RECORD EACH FLIP
• FIND HOW MANY “SAME’S” AND “DIFFERENT’S”
YOU GET
THE GAMBLER’S FALLACY
• WHAT DOES THE GAMBLERS FALLACY SAY
ABOUT THE ROULETTE STRATEGY?
• HOW IS THE GAMBLER’S FALLACY RELATED TO
STRATEGIES FOR THE GAME OF PIG...
THE GAMBLER’S FALLACY
…DID YOU SOMETIMES STOP ROLLING BECAUSE
YOU THOUGHT THERE WAS NOW A GREATER
CHANCE OF ROLLING A 1?
WHAT ARE THE CHANCES?
• REACHING INTO A BAG WITH 3 RED, 2 BLUE,
AND 4 BLACK GUM BALLS AND PULLING OUT A
BLUE?
• FLIPPING A COIN TWICE AND GETTING
DIFFERENT RESULTS?
WHAT ARE THE CHANCES?
• SNOW FALLING SOMETIME IN JULY IN FLORIDA
• SNOW FALLING SOMETIME IN JULY IN NEW
ZEALAND
WHAT ARE THE CHANCES?
• GOING TO A STORE AND NOT FINDING A TSHIRT IN YOUR SIZE.
• A STUDENT IN YOUR MATH CLASS WEARING
SNEAKERS.
WHAT ARE THE CHANCES?
• ROLLING A DIE AND GETTING A PRIME
NUMBER.
• ROLLING 2 DICE AND GETTING DOUBLES.
• ROLLING A PAIR OF DICE AND GETTING
DOULBES BY THE 3RD ROLL.
PAULA’S PIZZA
• WHAT IS THE PROBABILITY SHE WILL GET
WHAT SHE ORDERED?
• WHAT IS THE PROBABILITY SHE WILL NOT?
PAULA’S PIZZA
• PAULA ACTUALLY LIKES SAUSAGE AND
PEPPERONI TOO. WHAT IS THE PROBABILITY
SHE WILL GET A PIZZA SHE LIKES?
• AND THE PROBABILITY SHE DOESN’T?
RUG GAMES
WHAT’S THE
PROBABILITY OF BEING
HIT FOR EACH COLOR?
RUG GAMES
RUG GAMES
ROLLIN’, ROLLIN’, ROLLIN’
• ROLL A PAIR OF DICE 50 TIMES AND RECORD
THE SUMS
• DRAW A GRAPH OF THE DATA YOU GATHERED
ROLLIN’, ROLLIN’, ROLLIN’
• WHAT NEW THOUGHTS DOES THIS
EXPERIMENT GIVE YOU ABOUT HOW TO PLAY
THE COUNTERS GAME?
THEORY OF TWO DICE SUMS
TWO-DICE SUMS AND PRODUCTS
• WHICH IS MORE LIKELY--THAT THE SUM IS AN
ODD NUMBER OR THAT THE SUM IS AN EVEN
NUMBER?
• MAKE UP TWO PROBABILITY QUESTIONS
ABOUT 2 DICE SUMS.
TWO-DICE SUMS AND PRODUCTS
• WHAT ARE THE POSSIBLE TWO-DICE
PRODUCTS?
• WHAT IS THE PROBABILITY OF GETTING EACH
OF THE TWO-DICE PRODUCTS?
TWO-DICE SUMS AND PRODUCTS
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
2
4
6
8
10
12
3
6
9
12
15
18
4
8
12
16
20
24
5
10
15
20
25
30
6
12
18
24
30
36
TWO-DICE SUMS AND PRODUCTS
• WHICH IS MORE LIKELY--THAT THE TWO-DICE
PRODUCT IS AN ODD NUMBER OR THAT THE
TWO-DICE PRODUCT IS AN EVEN NUMBER?
SPINNER GIVE AND TAKE
• WRITE DOWN WHO YOU PREDICT TO BE THE
WINNER OF THIS GAME.
• NOW PLAY THE GAME 25 TIMES AND RECORD
YOUR RESULTS.
SPINNER GIVE AND TAKE
• IF AL AND BETTY PLAY 100 GAMES, HOW FAR
AHEAD IS THE EXPECTED WINNER LIKELY TO
BE?
MIA’S CARDS
• IF MIA PICKS A CARD MANY TIMES, WHAT WILL
BE HER AVERAGE NUMBER OF POINTS FOR
EACH TIME SHE PICKS A CARD?
MIA’S CARDS
• ON FEB. 14TH, SHE CHANGES THE GAME.
– 20 PTS FOR A HEART
– 15 PTS FOR A DIAMOND
– 0 PTS FOR A CLUB OR SPADE
• WHAT WOULD HER AVERAGE SCORE PER
CARD BE?
A FAIR RUG GAME?
• IS THIS A FAIR GAME?
• WHAT IS TONY’S EXPECTED VALUE FOR EACH
TURN?
• WHAT IS CRYSTAL’S?
A FAIR RUG GAME?
• IF YOU THINK THE GAME IS NOT FAIR, CHANGE
THE AMOUNT OF MONEY THEY EACH WIN IN
ORDER TO MAKE IT A FAIR GAME.
A SIXTY-PERCENT SOLUTION
• DO THE SIMULATION 40 TIMES
• WHAT WAS THE MOST FREQUENT OUTCOME?
• WHAT WAS THE AVERAGE SCORE?
THE THEORY OF ONE-AND-ONE
STREAK-SHOOTING SHELLY
• IN WHAT % OF SITUATIONS WILL SHELLY
SCORE NO POINTS?
• ONE POINT?
• TWO POINTS?
STREAK-SHOOTING SHELLY
• WHAT IS SHELLY’S EXPECTED VALUE PER ONEAND-ONE SITUTAION?
SPINS AND DRAWS
• AL’S WINS 1/5 OF THE TIME
• BETTY’S WINS 4/5 OF THE TIME
• AL WINS $1.25
• BETTY WINS 30 CENTS
SPINS AND DRAWS
• WHAT IS AL’S EXPECTED VALUE PER SPIN?
• WHAT IS BETTY’S EXPECTED VALUE PER SPIN?
SPINS AND DRAWS
• HOW MIGHT THE PAYMENTS BE CHANGED SO
THAT THE GAME IS FAIR?
SPINS AND DRAWS
• ARCHIBALD
– WINS 20 CENTS IF
• A JACK IS DRAWN
• BEATRICE
– WINS 8 CENTS IF
• A HEART IS DRAWN
• CHARITY
– WINS A PENNY IF
• NEITHER A JACK
OR HEART IS
DRAWN
MARTIAN BASKETBALL
• HOW MANY POINTS IS SHE MOST LIKELY TO
SCORE IN A ONE-AND-ONE SITUATION?
1ST SHOT 60 %
2ND SHOT 80%
3RD SHOT 90%
THE CARRIER’S PAYMENT PLAN
QUANDARY
THE CARRIER’S PAYMENT PLAN
QUANDARY
TWO POINT STRATEGY
TWO DRAW STRATEGY
THREE POINT STRATEGY
THREE DRAW STRATEGY
LITTLE PIG
POINTS
VS
DRAWS
1 point
1.6
1 draw
2 point
2 draw
3 point
3 draw
4 point
4 draw
5 point
5 draw
6 point
6 draw
7 point
1.6
2.1
2.2
2.2
2.2
2.247
1.975
2.362
1.646
2.362
1.317
2.325
LITTLE PIG
POINTS VS DRAWS
1 draw
2 draw
3 draw
4 draw
5 draw
6 draw
1.6
2.2
2.2
1.975
1.646
1.317
1 point
2 point
3 point
4 point
5 point
6 point
7 point
1.6
2.1
2.2
2.247
2.362
2.362
2.325
THE BEST LITTLE PIG
• BASED ON YOUR FINDINGS, WHAT STRATEGY
WILL GIVE THE HIGHEST POSSIBLE EXPECTED
VALUE PER TURN?