1
Coin Tossing Games
1) Game 1 of tossing one coin.
Keep tossing a coin until A or B wins. A wins on contiguous HH, and B wins on HT.
Q: [ ] [ ]?P Awins P Bwins= .
2) Game 2 of tossing one coin.
Keep tossing a coin until A or B wins. A wins on contiguous HH, and B wins on TH.
Q: [ ] [ ]?P Awins P Bwins=
3) Game of tossing own coin.
Each player has their own coin. A wins on contiguous HH, and B wins on HT.
Q: [ ] [ ]?P Awins P Bwins=
Solution of Game 3
Define
( ) : probability gets for the first time at the -th toss, 1,2,
( ) : probability gets for the first time at the -th toss, 1,2,A
B
P j A HH j j
P j B HT j j
==
[ ]
1 1
( ) ( ) and ( ) ( )
Then
( ) gets at or before the -th toss
( ) ( ) ( 1)
k k
A A B Bj j
A
A A A
Q k P j Q k P j
Q k P A HH k
P k Q k Q k
= =
== − −
[ ]Likewise
( ) gets at or before the -th toss
( ) ( ) ( 1)B
B B B
Q k P B HT k
P k Q k Q k
== − −
[ ] { } { }
[ ] { }
[ ]
1 1
1
1
Once ( ) and ( ) are found, the winning probabilities can be found from
( ) ( 1) ( 2) ( ) 1 ( )
( ) 1 ( )
( ) ( )
A B
A B B A Bk k
B Ak
A Bk
Q k Q k
P A wins P k P k P k P k Q k
P B wins P k Q k
P tie P k P k
∞ ∞
= =∞
=∞
=
= + + + + = −
= −
=
2
Find ( ) for 1, 2,3, 4.AP j j =
A general value of ( ) is difficult to find. We will find 1 ( ) instead.A AP k Q k−
( )paths to be counted for 2AP
H
T
( )paths to be counted for 3AP
H
T
( )paths to be counted for 4AP
H
T
3
Suppose has never got until the -th toss.
Define
( ) : the number of such sequences that end with a .
( ) : the number of such sequences that end with a .
Then
( ) ( )1 ( )
2
HA
TA
H TA A
A k
A HH k
u k Head
u k Tail
u k u kQ k
+− =
Likewise, suppose has never got until the -th toss.
Define
( ) : the number of such sequences that end with a .
( ) : the number of such sequences that end with a .
Then
( )1 ( )
HB
TB
H TB B
B
B HT k
u k Head
u k Tail
u k uQ k
+− = ( )
2kk
0222 2k
H
T
1 tossst 2nd
at least one 'HH' along the path
tossthk
no 'HH' along the pathending with 'H'
no 'HH' along the pathending with 'T'
( ) is the probability of a blue path.AQ kLeaf nodes form a simple sample space.
( )1 is the probability of a red path.AQ k−
4
Study of A, no HH up to the k-th toss
(1) { } 1 (1) { } 1
(2) { } 1 (2) { , } 2
(3) (2) (3) (2) (2)
( ) ( 1) for 2,3,
( ) ( 1) ( 1)
( 1) ( 2) for 3,4,
H TA A
H TA A
H T T H TA A A A A
H TA A
T T HA A A
T TA A
u H u T
u TH u HT TT
u u u u u
In general
u k u k k
u k u k u k
u k u k k
= = = =
= = = =
= = +
= − =
= − + −
= − + − =
Such a sequence is called the Fibonacci sequence.
1 11 1 5 1 5
( )2 25
k kTAu k
+ + + − = −
2 ( ) ( )( )
2
( ) ( ) ( 1)
k H TA A
A k
A A A
u k u kQ k
P k Q k Q k
− −=
= − −
Self Study.
{ }0 1
1 2
Derive the Fibonacci sequence 1,1,2,3,5,8, .
1.
for 2,3,n n n
x x
x x x n− −
= =
= + =
5
Study of B, no HT up to the k-th toss
(1) 1 (1) 1{ } { }
(2) 2 (2) 1{ , } { }
In general,
( ) ( ) 1
H TB B
H TB B
H TB B
u uH T
u uHH TH TT
u k k u k
= = = =
= = = =
= =
2 ( ) ( )( )
2
( ) ( ) ( 1)
k H TB B
B k
B B B
u k u kQ k
P k Q k Q k
− −=
= − −
Putting all together
[ ] { }1
39( ) 1 ( )
121A Bk
P A wins P k Q k∞
== − =
[ ] { }1
65( ) 1 ( )
121B Ak
P B wins P k Q k∞
== − =
[ ]
1
17( ) ( )
121A Bk
P tie P k P k∞
== =
6
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 2 4 6 8 10 12
k
P[A win]
P[B win]
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