Mathematics for Computer Science
MIT 6.042J/18.062J
Gambler’s Ruin
Albert R Meyer,
December 14,
Lec 15W.1
Gambler’s Ruin
• Place $1 bets until going
broke or reaching target
• What is Pr[reach target]?
Albert R Meyer,
December 14,
Lec 15W.3
Gambler’s Ruin
T ”target"
$
$
n "initial capital"
# of bets
Albert R Meyer,
December 14,
Lec 15W.4
Dow Jones Trend
random steps with “up” bias?
Albert R Meyer,
December 14,
Lec 15W.5
Gambling: Fair Case
Suppose we’re playing a fair game:
• Pr[win bet] = 1/2.
What is Pr[reach $200] if we start
with $100?
1/2
What about Pr[reach $600] if we
start with $500?
5/6
Albert R Meyer,
December 14,
Lec 15W.6
Gambling: Fair Case
For fair game in
general
What about an unfair game?
Albert R Meyer,
December 14,
Lec 15W.7
Gambling: Slightly Unfair
Betting black in
US roulette
Pr[win bet] = 18/38 = 9/19 < 1/2
Albert R Meyer,
December 14,
Lec 15W.8
US Roulette
What is Pr[reach $500+100] starting
with $500?
(5/6 when fair)
< 1 / 37,000
What is Pr[reach $1,000,100] starting
with $1,000,000? (≈ 1 when fair)
< 1 / 37,000
no matter how many $ at start!
Albert R Meyer,
December 14,
Lec 15W.9
Gambler’s Ruin
Parameters
p ::= Pr[win $1 bet]
n ::= initial capital
T ::= gambler’s target
What is Pr[reach target]?
Albert R Meyer,
December 14,
Lec 15W.10
Condition on 1st bet
wn::= Pr[win starting with $n]
st
Pr[wn| win 1 bet] = wn+1
st
Pr[wn| lose 1 bet] = wn-1
wn= wn+1∙p + wn-1∙q
Albert R Meyer,
December 14, 2011
Lec 15W.18
A Linear Recurrence
wn+1 = (1/p)wn - (q/p)wn-1
w0 = 0
wT = 1
(Gambler is broke)
(Gambler at target)
Solve as usual and get:
Albert R Meyer,
December 14,
Lec 15W.19
Linear Recurrence
Twist: we
don’t know w1
for r ::= q/p ≠ 1.
But wT = 1, so can solve for w1. Then
Albert R Meyer,
December 14,
Lec 15W.20
Winning when Biased Against
intended profit
Suppose p < q, so r ::= q/p
> 1.
Albert R Meyer,
December 14,
Lec 15W.21
Winning when Biased Against
wn <
intended
profit
(1/r)
wn bound does not depend on n!
1/r < 1, so wn is exponentially
decreasing in intended profit!
Albert R Meyer,
December 14,
Lec 15W.22
Profit $100 in US Roulette
p = 18/38
q = 20/38
1/r = 9/10
Pr[Profit $100] <
100
(9/10)
< 1/37,648
Albert R Meyer,
December 14,
Lec 15W.23
Profit $200 in US Roulette
p = 18/38
q = 20/38
1/r = 9/10
Pr[Profit $200] <
200
(9/10)
<
2
(1/37,648)
< 1/70,000,000
Albert R Meyer,
December 14,
Lec 15W.24
What About the Fair Case?
(r ::= q/p = 1)
Uh oh, dividing by 0.
Use l’Hôpital’s Rule
d(rn-1)/dr nrn-1
limr→1
=
=
d(rT-1)/dr TrT-1
Albert R Meyer,
December 14,
n
T
Lec 15W.25
Expected number of bets
Albert R Meyer,
December 14, 2011
Lec 15W.26
Apply Total Expectation
Albert R Meyer,
December 14, 2011
Lec 15W.27
Linear recurrence
Albert R Meyer,
December 14, 2011
Lec 15W.28
Expected number of bets
Solve linear recurrence as usual.
Elegant result in the fair case:
en = n(T-n)
= (initial stake)
∙(intended profit)
Albert R Meyer,
December 14, 2011
Lec 15W.29
Expected number of fair bets
For example, starting with $1
aiming to reach $1000,
expect to make 999 bets
(and most likely go broke)
Problem: There must be an
intuitive proof. Find one.
Albert R Meyer,
December 14, 2011
Lec 15W.30
Team Problems
Problems
1&2
Albert R Meyer,
December 14,
Lec 15W.31