6. Monopoly Rules
•
•
•
•
•
40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
7. Monopoly Rules
•
•
•
•
•
40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
8. Monopoly Rules
•
•
•
•
•
40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
9. Monopoly Rules
•
•
•
•
•
40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
10. Monopoly Rules
•
•
•
•
•
40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
11. Monopoly Rules
•
•
•
•
•
40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
12. Monopoly Rules
•
•
•
•
•
40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
13. Monopoly Rules
•
•
•
•
•
40 4 spaces (Go through Boardwalk)
Roll two six-sided dice Flip a coin to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
14. Suppose we only have four
spaces (A, B, C, and D) and
that a move consists of
flipping a coin.
• Heads = Move two
spaces
• Tails = Move one space
26. Monopoly: Terminally Boring Edition
•
•
•
•
•
40 4 spaces (Go through Boardwalk)
Roll two six-sided dice Flip a coin to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
27. Monopoly: Simple Model
•
•
•
•
•
40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
28. Rolling Two Six-Sided Dice
Spaces Moved
Probability
2
1/36
3
2/36
4
3/36
5
4/36
6
5/36
7
6/36
8
5/36
9
4/36
10
3/36
11
2/36
12
1/36
43. Markov Chains
Definition: A vector with the
property that the sum of its
entries is 1 is called a probability
vector.
Definition: A square matrix with
the property that the sum of the
entries in each of its columns is 1
is called a stochastic matrix.
Andrey Markov, 1856 – 1922
44. Markov Chains
Definition: A Markov chain is a dynamical system for
which
• the probability vector xk describes the state of the
system at time k and
• successive state vectors are related by the following
equation, where P is a stochastic matrix called the
transition matrix for the system.
xk+1=Pxk
45. Markov Chains
Theorem: If P is the transition matrix for a Markov
chain (and P is regular), then…
• There is a unique probability vector q such that
Pq=q.
• For any initial state vector x0,
xk q as k
Finding q means solving the equation
Pq=q
47. Monopoly: Model #2
•
•
•
•
•
40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
50. Monopoly: Model #3
•
•
•
•
•
40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
52. Monopoly: Model #4
•
•
•
•
•
40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail.
Rolling three doubles in a row sends you to Jail.
Get out of jail by…
–
–
–
–
Paying $50,
Using a “Get out of Jail, Free” card,
Rolling doubles, or
Spending three turns in Jail.
• Chance and Community Chest cards have various
effects.
54. What’s Left?
• Rolling three doubles in a row sends you to
Jail.
• Chance and Community Chest cards have
various effects.
You still have two underlying models—leave jail
quickly or stay as long as you can.
68. Flickr Credits
• “monopoly,” foreverdigital
• “Black Dice,” Mariano Kamp
• “Monopoly,” unloveablesteve
• “last man standing,” Robert Terrell
• “Racing for Home,” Scott Ableman
• “Nomads (brog pa) crossing Lha chu at Kailash
Kora,” reurinkjan
Notas do Editor
“monopoly,” by Flickr user foreverdigital, http://www.flickr.com/photos/foreverdigital/4159039717/
“Black Dice,” by Flickr user Mariano Kamp, http://www.flickr.com/photos/mkamp/2478311790/
“Monopoly,” by Flickr user unloveablesteve, http://www.flickr.com/photos/unloveable/2400877902/
Strategic LearnersThese learners react well to competition and the chance to do better than anyone else.They often become strategic learners, making high grades but seldom grappling deeply enough to change their own perceptions.They are often “regurgitators”—learning material for the test and then quickly expunging the material to make room for something else.
“Racing for Home,” by Flickr user Scott Ableman, http://www.flickr.com/photos/ableman/183059002/
“Nomads (brog pa) crossing Lhachu at KailashKora,” by Flickr user reurinkjan, http://www.flickr.com/photos/reurinkjan/3127359433/
Mathematical PageRanks (out of 100) for a simple network (PageRanks reported by Google are rescaled logarithmically). Page C has a higher PageRank than Page E, even though it has fewer links to it; the link it has is of a much higher value. A web surfer who chooses a random link on every page (but with 15% likelihood jumps to a random page on the whole web) is going to be on Page E for 8.1% of the time. (The 15% likelihood of jumping to an arbitrary page corresponds to a damping factor of 85%.) Without damping, all web surfers would eventually end up on Pages A, B, or C, and all other pages would have PageRank zero. Page A is assumed to link to all pages in the web, because it has no outgoing links.http://en.wikipedia.org/wiki/PageRank