Mathematics for Computer Science
Graph with probable transitions
MIT 6.042J/18.062J
4/7
1/6
Random Walks
4/9
1/6
1/7
2/7
3/5
2/3
5/9
2/5
Copyright ©Albert R. Meyer, 2005.
December 14, 2005
lec 15w.1
Copyright ©Albert R. Meyer, 2005.
Graph with probable transitions
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Applications
• Finance – Stocks, options
• Algorithms – web search, clustering
• Physics – Brownian Motion
Copyright ©Albert R. Meyer, 2005.
1-Dimensional Walk
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Gambler’s Ruin
"
"
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Random Walks
Questions
• Pr{blue reaches orange before green}
• Pr{blue ever reaches orange}
• E[#steps blue to orange]
• Average fraction of time at blue
Copyright ©Albert R. Meyer, 2005.
December 14, 2005
p
T
$$$
q
Gambler’s Ruin
n
0
Copyright ©Albert R. Meyer, 2005.
December 14, 2005
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Copyright ©Albert R. Meyer, 2005.
# of bets
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1
Gambler’s Ruin
Gambler’s Ruin
Parameters:
Three general cases:
n ::= initial capital (stake)
T ::= gambler’s Target
p ::= Pr{win $1 bet}
q ::=1– p
m ::= intended profit = T – n
Copyright ©Albert R. Meyer, 2005.
• Biased against
• Biased in favor
• Unbiased (Fair)
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Fair Case: p = q = 1/2
Copyright ©Albert R. Meyer, 2005.
Let w ::= Pr{reach Target}
E[$$] = w·(T – n) + (1 – w)·(–n)
= wT – n
But game is fair, so E[$$ won] = 0
Copyright ©Albert R. Meyer, 2005.
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December 14, 2005
Copyright ©Albert R. Meyer, 2005.
December 14, 2005
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Biased Against: p < 1/2 < q
Consequences
n=500, T=600
Pr{win $100} = 500/600 ≈ 0.83
Betting red in US roulette
p = 18/38 = 9/19 < 1/2
n=1,000,000, T=1,000,100
Pr{win $100} ≈ 0.9999
December 14, 2005
n
w=
T
n
T
Fair Case
Copyright ©Albert R. Meyer, 2005.
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December 14, 2005
Fair Case: p = q = 1/2
Let w ::= Pr{reach Target}
w=
p < 1/2
p > 1/2
p = 1/2
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Copyright ©Albert R. Meyer, 2005.
December 14, 2005
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2
Biased Against: p < 1/2 < q
Biased Against: p < 1/2 < q
Astonishing Fact!
Pr{win $100 starting with $500}
< 1/37,000 !
More amazing still!
Pr{win $100 starting with $1M}
< 1/37,000
Pr{win $100 starting w/ any $n stake}
(was 5/6 in the unbiased case.)
Copyright ©Albert R. Meyer, 2005.
December 14, 2005
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Winning in the Unfair Case
< 1/37,000
Copyright ©Albert R. Meyer, 2005.
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Winning in the Unfair Case
Team Problem: for p < q,
for p < q:
⎛ p⎞
⎜ ⎟
⎝q⎠
m
is exponentially decreasing in m,
where m ::= T−n = intended profit
Copyright ©Albert R. Meyer, 2005.
December 14, 2005
the intended profit.
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Copyright ©Albert R. Meyer, 2005.
Losing in Roulette
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Losing in Roulette
p = 18/38, q = 20/38
Copyright ©Albert R. Meyer, 2005.
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Copyright ©Albert R. Meyer, 2005.
December 14, 2005
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3
Gambler’s Ruin
Gambler’s Ruin
T
T
T-n
$$$
$$$
n
0
0
# of bets
Copyright ©Albert R. Meyer, 2005.
December 14, 2005
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pr{lose starting with $n}
= pr{win starting with $(T - n)}
T −n
T
Copyright ©Albert R. Meyer, 2005.
December 14, 2005
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Fair Case for T = ∞
Fair Case
=
# of bets
Copyright ©Albert R. Meyer, 2005.
Pr{lose starting with n | T = ∞}
≥ Pr{lose starting with n | T < ∞}
T −n
→∞
as T → ∞
=
T
So if the gambler keeps betting,
he is sure to go broke.
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Copyright ©Albert R. Meyer, 2005.
December 14, 2005
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How Many Bets?
Return to the origin.
What is the expected number of
bets for the game to end?
– either by winning $(T-n) or by going broke (losing $n).
If you start at the origin and move left or right with equal probability, and keep moving in this way, Pr{return to origin} = 1
Copyright ©Albert R. Meyer, 2005.
December 14, 2005
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Copyright ©Albert R. Meyer, 2005.
December 14, 2005
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4
How Many Bets? Fair Case
Fair Case for T = ∞
E[# bets] = n(T−n) =
(initial stake)⋅(intended profit)
proof by solving linear recurrence:
en = p(1 + en+1) + q(1 + en-1)
Copyright ©Albert R. Meyer, 2005.
December 14, 2005
lec 15w.30
Likewise,
E[#bets for T = ∞]
≥ E[#bets for T < ∞]
= n(T-n) → ∞ (as T → ∞)
So the expected #bets to go broke is
infinite!
Copyright ©Albert R. Meyer, 2005.
December 14, 2005
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Team Problems
Problems 1−3
Copyright ©Albert R. Meyer, 2005.
December 14, 2005
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5