1 . This question aims at investigating a coin tossing game by probability models and graph theory .
A coin tossing game consists of a coin with two faces : head and tail , where the probability of getting a head is p . The player of this game keeps tossing the coin and the results are recorded . The game stops if the player gets three consecutive heads . If a tail is shown , the player will restart the game .
Let
1 p � . Four states are used to describe the game :
2
A : The player starts the game
B : One head has shown
C : Two consecutive heads have shown
D : Three consecutive heads have shown
( a ) Sketch a directed graph to represent the flow of this game . [ 4 ]
( b ) State the geometric meaning of the column sum of a particular column in an adjacency matrix of a graph .
[ 1 ]
( c ) Write down
( i ) the adjacency matrix M of the graph ;
( ii ) the transition matrix T of the graph .
Let v n be the state probability vector after the coin is tossed for n times .
[ 4 ]
�1� � � 0
( d ) ( i ) Explain why v
0
� � �. � 0 � � � �0�
( ii ) Hence , write down v
1 and v
2
.
��1
� � � �2
It is given that v
3
� � �. � �3 � � � ��4
�
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