Probability Calculations and Strategies in Liars Dice Tournament
Agenda
•
Warm Up
10 min
•
Liar’s Dice tournament
45 min
–
Set up Excel
–
Update w/ math, make worksheet
–
Make groups
•
Project time?
•
Exit Pass
5 min
Homework (reg)
Casino Project calculations due tomorrow
Warm Up
Engineers define reliability as the probability that a
machine will perform its function. A certain model of
aircraft engine is designed so that each engine has
probability 0.999 of performing properly. Engineers test
an SRS of 350 engines of this model.
1.
Find the probability that all the engines perform
properly.
2.
Two engines fail the test. Are you convinced that this
model of engine is unreliable? Justify with
appropriate probability calculations.
(Hint: Find P(x ≤ 348)
Liar’s Dice
Liar’s Dice, rules
Roll dice
(hide from opponents).
Loser decides who starts
(may choose self)
Players take turns. On your turn, you can….
1.
Raise bet.
Must say a higher bet.
•
Bets are always “at least” bets.
•
Bets are based on all dice on table.
•
1’s are “wild” (everything), unless someone bets 1’s.
2.
“Challenge”.
(can be said
any time)
•
All players reveal dice. Wrong person loses one die.
3.
“Exactly”.
•
All players reveal dice. Must be
exactly
that amount, no more, no less. If
correct, all other players lose one die each.
And repeat! Re-roll, start over. Winner = last player with dice.
Liar’s Dice, math
Use binomial to win!
EXAMPLE:
•
I’m playing with 3 other people (16 dice total).
•
I have four dice, and three of them are 6’s or 1’s.
•
I bet “seven 6’s”
•
What are the chances I’m right?
•
There are 12 more dice.
•
I need four more 6’s or 1’s.
Three or less….I lose.
More examples
There are 13 dice left among all players, including me.
I have 5 dice left, so the other players have 8 dice.
Three of my dice are 5’s or 1’s.
1.
If I bet “five 5’s”, what is the probability that I am right?
2.
If I bet “seven 5’s”, what is the probability that I am right?
3.
If I bet “ten 5’s “, what is the probability that I am right?
Try it!
½ worksheet
1.
There are 15 dice left. You have 4
dice, with three 6’s. If you bet “six
6’s”, what is the probability that
you are right?
2.
There are 10 dice left. You have 3
dice, with two 4’s. If you bet “five
4’s”, what is the probability that
you are right?
3.
There are 18 dice left. You have 5
dice, with three 6’s. If you bet “six
6’s”, what is the probability that
you are right?
4.
There are 16 dice left.
But watch
out, 1’s have been killed!
You have
5 dice, with three 5’s. If you bet
“six 5’s”, what is the probability
that you are right?
Round 2
1.
Kayla, Claudia, Camille, Daniel
2.
Josh, Jamie, Aleftina, Syed
3.
Anthony, Spencer, Karina, Beloved
4.
Joey, Hayley, Melissa, Ally
5.
Taylor, Ari, Shiny, Darya
6.
Riya, Lissette, Gurpreet
Period 2
Round 3
1.
Ally, Beloved, Gurpreet, Kayla
2.
Daniel, Shiny, Karina, Darya
3.
Syed, Melissa, Aleftina, Lissette
4.
Camille, Ari, Spencer, Hayley
5.
Jamie, Riya, Claudia, Taylor
6.
Anthony, Joey, Josh
Period 2
Casino Project
Create, mathematically analyze, and run a simple gambling game which is
significantly (but not obviously) in favor of you, “The House”.
1-2 people (2 recommended)
Up to +5% extra credit for binomial probabilities and/or confidence intervals.
Wednesday March 6
th
, due end of class:
1.
List three possible ideas for your project. For each game, state how to play and
how to win.
Thursday March 14
th
(or earlier), due end of class:
2.
Choose one game and give it a catchy name. Describe your game (including the
title), and calculate its probabilities and expected value.
Wednesday March 20
th
:
3.
Bring an attractive poster that explains your game, including prices and prizes.
Bring all components needed for your game.
Monday March 25
th
: CASINO DAY!
7.
Run your game, attract gamblers, and help them lose their money! You must
profit at least $500 ($1000 total).
Exit Pass
A psychologist studied the number of puzzles that subjects
were able to solve in a five-minute period while listening to
soothing music. This is the probability distribution for
X
number of puzzles.
1.
What is the expected number of solved puzzles?
2.
Suppose that three randomly selected subjects solve
puzzles for five minutes each. What is the expected
value of the total number of puzzles solved by the
three subjects?
"Explore the world of probabilities and strategies in a Liars Dice tournament, from calculating engine reliability to using binomial distributions to win bets. Learn how to analyze dice rolls, place strategic bets, and challenge opponents to emerge as the ultimate winner in this thrilling game! Join the fun and sharpen your mathematical skills along the way."
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Presentation Transcript
Homework (reg) Agenda Casino Project calculations due tomorrow Warm Up Liar s Dice tournament Set up Excel Update w/ math, make worksheet Make groups Project time? Exit Pass 10 min 45 min 5 min
Warm Up Engineers define reliability as the probability that a machine will perform its function. A certain model of aircraft engine is designed so that each engine has probability 0.999 of performing properly. Engineers test an SRS of 350 engines of this model. 1. Find the probability that all the engines perform properly. 2. Two engines fail the test. Are you convinced that this model of engine is unreliable? Justify with appropriate probability calculations. (Hint: Find P(x 348)
Liars Dice, rules Roll dice (hide from opponents). Loser decides who starts (may choose self) Players take turns. On your turn, you can . 1. Raise bet. Must say a higher bet. Bets are always at least bets. Bets are based on all dice on table. 1 s are wild (everything), unless someone bets 1 s. 2. Challenge . (can be said any time) All players reveal dice. Wrong person loses one die. 3. Exactly . All players reveal dice. Must be exactly that amount, no more, no less. If correct, all other players lose one die each. And repeat! Re-roll, start over. Winner = last player with dice.
Liars Dice, math Use binomial to win! EXAMPLE: I m playing with 3 other people (16 dice total). I have four dice, and three of them are 6 s or 1 s. I bet seven 6 s What are the chances I m right? There are 12 more dice. I need four more 6 s or 1 s. Three or less .I lose. 2 binomcdf = 1 12 ( , ) 3 , 6 . 0 607
More examples There are 13 dice left among all players, including me. I have 5 dice left, so the other players have 8 dice. Three of my dice are 5 s or 1 s. 1. If I bet five 5 s , what is the probability that I am right? 2 binomcdf = 1 , 8 ( ) 1 , 6 8049 . 0 2. If I bet seven 5 s , what is the probability that I am right? 2 binomcdf = 1 , 8 ( ) 3 , 6 2587 . 0 3. If I bet ten 5 s , what is the probability that I am right? 2 binomcdf = 1 , 8 ( ) 6 , 6 0026 . 0
Try it! worksheet
1. There are 15 dice left. You have 4 dice, with three 6 s. If you bet six 6 s , what is the probability that you are right? 2 binomcdf = 1 11 ( , ) 2 , 6 . 0 766 2. There are 10 dice left. You have 3 dice, with two 4 s. If you bet five 4 s , what is the probability that you are right? 2 binomcdf = 1 , 7 ( ) 2 , 6 4294 . 0 3. There are 18 dice left. You have 5 dice, with three 6 s. If you bet six 6 s , what is the probability that you are right? 2 binomcdf = 1 13 ( , ) 2 , 6 . 0 861 4. There are 16 dice left. But watch out, 1 s have been killed! You have 5 dice, with three 5 s. If you bet six 5 s , what is the probability that you are right? 1 binomcdf = 1 11 ( , ) 2 , 6 . 0 273
Period 2 Round 2 1. Kayla, Claudia, Camille, Daniel 2. Josh, Jamie, Aleftina, Syed 3. Anthony, Spencer, Karina, Beloved 4. Joey, Hayley, Melissa, Ally 5. Taylor, Ari, Shiny, Darya 6. Riya, Lissette, Gurpreet
Period 2 Round 3 1. Ally, Beloved, Gurpreet, Kayla 2. Daniel, Shiny, Karina, Darya 3. Syed, Melissa, Aleftina, Lissette 4. Camille, Ari, Spencer, Hayley 5. Jamie, Riya, Claudia, Taylor 6. Anthony, Joey, Josh
Casino Project Create, mathematically analyze, and run a simple gambling game which is significantly (but not obviously) in favor of you, The House . 1-2 people (2 recommended) Up to +5% extra credit for binomial probabilities and/or confidence intervals. Wednesday March 6th, due end of class: 1. List three possible ideas for your project. For each game, state how to play and how to win. Thursday March 14th (or earlier), due end of class: 2. Choose one game and give it a catchy name. Describe your game (including the title), and calculate its probabilities and expected value. Wednesday March 20th: 3. Bring an attractive poster that explains your game, including prices and prizes. Bring all components needed for your game. Monday March 25th: CASINO DAY! 7. Run your game, attract gamblers, and help them lose their money! You must profit at least $500 ($1000 total).
Exit Pass A psychologist studied the number of puzzles that subjects were able to solve in a five-minute period while listening to soothing music. This is the probability distribution for X number of puzzles. Value of X 1 2 3 4 Prob. 0.2 0.4 0.3 0.1 1. What is the expected number of solved puzzles? 2. Suppose that three randomly selected subjects solve puzzles for five minutes each. What is the expected value of the total number of puzzles solved by the three subjects?
