Dice Games: Probability and Pascal by Lauren McCluskey

1
Dice Games Probability and Pascal by Lauren
McCluskey

2

• This power point was made with help from
• Bayesian Learning Application to Text
  Classification Example spam filtering by Marius
  Bulacu prof. dr. Lambert Schomaker
• Mathematicians by www.2july-maths.co.uk/powerpoint
  /mathematicians.ppt
• Basic Models of Probability by Ron S. Kenett,
  Weizmann Institute of Science Probability
• Introduction to Information Theory by Larry
  Yaeger, Professor of Informatics, Indiana
  University
• Access to Math, Probability published by
  www.pearsonlearning.com
• www.2july-maths.co.uk/powerpoint/mathematicians.pp
  t
• www.mtsu32.mtsu.edu11208/Chap9Pres.ppt

3
Founders of Probability Theory
Blaise Pascal (1623-1662, France)
Pierre Fermat (1601-1665, France)
They laid the foundations of the probability
theory in a correspondence on a dice game. From
Bayesian Learning Application to Text
Classification Example spam filtering by Marius
Bulacu prof. dr. Lambert Schomaker
4
Pascal from Mathematicians by www.2july-maths.co.
uk/powerpoint/mathematicians.ppt

• Blaise Pascal
• 1623 - 1662
•   
• Blaise Pascal, according to contemporary
  observers, suffered migraines in his youth,
  deplorable health as an adult, and lived much of
  his brief life of 39 years in pain.
• Nevertheless, he managed to make considerable
  contributions in his fields of interest,
  mathematics and physics, aided by keen curiosity
  and penetrating analytical ability.

5
Pascal from Mathematicians by www.2july-maths.co.
uk/powerpoint/mathematicians.ppt

• Probability theory was Pascal's principal and
  perhaps most enduring contribution to
  mathematics, the foundations of probability
  theory established in a long exchange of letters
  between Pascal and fellow French mathematician
  Fermat.  

6
Basic Models of Probability by Ron S. Kenett,
Weizmann Institute of Science
The Paradox of the Chevalier de Mere - 1
Game A
Success at least one 1
7
Basic Models of Probability by Ron S. Kenett,
Weizmann Institute of Science
The Paradox of the Chevalier de Mere - 2
Game B
Success at least one 1,1
8
Basic Models of Probability by Ron S. Kenett,
Weizmann Institute of Science
The Paradox of the Chevalier de Mere - 3
Game B
Game A
P (Success) P(at least one 1)
P (Success) P(at least one 1,1)
Experience proved otherwise !
Game A was a better game to play
9
Basic Models of Probability by Ron S. Kenett,
Weizmann Institute of Science
The Paradox of the Chevalier de Mere - 4
The calculations of Pascal and Fermat
Game A
Game B
P (Failure) P(no 1)
P (Failure) P(no 1,1)
P (Success) .518
P (Success) .491
What went wrong before?
10
Sample Space for Dice from Introduction to
Information Theory by Larry Yaeger

• Single die has six elementary outcomes
• Two dice have 36 elementary outcomes

11
1/61/61/61/6 (1/6)4 or .518

• While

12
1/361/361/36 (1/36)24 or .491
1/36 chances
13
Apply it!

• When to sit and when to stand?
• How many times can we roll one die before we get
  a 1?
• Try this

14
S.K.U.N.K.

• Stand up.
• 2. Someone rolls a die.
• 3. Sit down Keep your score.
• OR
• Remain standing Add it up.
• But

15
Watch Out!

• 4. If youre standing on 1 Score 0!
• 5. New round Stand up.
• 6. Repeat 5 times one round for each letter in
  the word S.K.U.N.K.

16
Reflection

• What is your winning strategy?
• Why will this work?
• Remember

17
Sample Space for Dice from Introduction to
Information Theory by Larry Yaeger

• Single die has six elementary outcomes
• Two dice have 36 elementary outcomes

18
Apply It!

• When to roll and when to stop?
• How many times can we roll 2 dice before we roll
  a 1 or a 1, 1?
• Try this

19
PIG

• Take turns rolling 2 dice.
• Keep rolling Add it up.
• Stop Keep your score.
• But

20
Watch Out!

• 4. Roll 1 Lose your turn.
• Roll 1, 1 Lose it ALL! (Back to 0!)
• 5. Get a score of 100 You WIN!

21
Reflection

• What is your winning strategy?
• Why will this work?
• Remember

22
Sample Space for Dice from Introduction to
Information Theory by Larry Yaeger

• Single die has six elementary outcomes
• Two dice have 36 elementary outcomes

1/36 chances
23
Sample Space for Dice from Introduction to
Information Theory by Larry Yaeger

• Single die has six elementary outcomes
• Two dice have 36 elementary outcomes

11/36 Chances To roll 1 1
24
The Addition Rule
from Introduction to Information Theory by Larry
Yaeger

• Now throw a pair of black white dice, and ask
  What is the probability of throwing at least one
  one?
• Let event a the white die will show a one
• Let event b the black die will show a one

25
Basic Models of Probability by Ron S. Kenett,
Weizmann Institute of Science
P(1 with 2 dice) ?
To add or to multiply ?
26
Independent Events

• Independent events two events with outcomes that
  do not depend on each other. (from Access to
  Math, Probability)

27
Independent Events Either /OR

• When two events are independent, AND either one
  is favorable, you add their probabilities.
• Example
• What is the probability that I might roll a 1 on
  the black die? 6/36 or 1/6
• What is the probability that I might roll a 1 on
  the white die? 6/36 or 1/6
• What is the probability that I will roll either 1
  black OR 1 white 1? 12/36 or 1/3.

28
Independent Events Either /OR

• This is true when the die are rolled one at a
  time, if, however, you roll them together, then
  1W and 1B cannot be counted twice. So the
  probability of rolling a 1 is 11/36 instead of
  12/36.

29
Sample Space for Dice from Introduction to
Information Theory by Larry Yaeger

• Single die has six elementary outcomes
• Two dice have 36 elementary outcomes

11/36 Chances To roll 1 1
30
Independent Events And Then

• When two events are independent, BUT you want to
  have BOTH of them, you multiply their
  probabilities.
• Example
• What is the probability that I will roll a
  1, 1? P(1) 1/6 P(1) 1/6 or 1/61/6
  1/36.
• The P(1,1) 1/36 because there is only ONE way
  that I can do this.

31
1/361/361/36 (1/36)24 or .491
1/36 chances
32
Dependent Events

• Dependent events a set of events in which the
  outcome of the first event affects the outcome of
  the next event.
• (from Access to Math, Probability)

33
Dependent Events

• To find the probability of dependent events,
  multiply the probability of the first by the
  probability of the second (given that the first
  has occurred).
• Example You have the letters M A T and H in
  an envelope. What is the probability that you
  will pull a M then a A?

34
Dependent Events

• P(M) ¼ because there are 4 cards and
• P(A after M) 1/3 because there are NOW only 3
  cards left
• so
• ¼ 1/3 1/12.

35
Apply It!

• Put the letters M A T and H in an envelope
  and pull them out 1 at a time.
• Replace the card then do it again.
• (Repeat 20 times.)
• Record your results.
• Think about it what would happen if you hadnt
  replaced the cards each time?