�

����Pn � pPn�� � qPn��

Arguing as in ����� we get the corresponding iteration equation

Pn � Pn�� � qPn��

and proceed as in Example �����

���� Suppose one best on k � �� �� � � � � ��Then

p� � P �k appears on one dice ����

� ���

� ���

��p� � P �k appear on two dice �

���

� ���

�� ���

�

p� � P �k appear on all the tree dice ����

��p� � P �k appear none �

���

��

Thus we get

Net gain � �p� � �p� � �p� � p� � �����

�

which gives for i � a

Na �

���������

a� bp� q �

�� �q�pa

� � �q�pa�b �a

p� q � p �� q

ab� p � q

�

���������

b�p� � � a� b

�p� � ��� �p�qb

�� �p�qa�b � p �� q�

ab� p � q

�

��� Using the hint we obtain

p �Nk�� �Nk � q �Nk �Nk��� �

Let

Mk�� � Nk�� �Nk

so that the above iteration gives

Mk�� � qp Mk �

�p

�

���������

�qp

�kM� �

�p � q

n�� �qp

ko� p �� q

M� �kp � p � q

This gives

Ni �i��Xk��

Mk��

�

�������������

�M� �

�p� q

� i��Xk��

�qp

�k� i

p � q � p �� q

iM� �i�i� �

�p � p � q

where we have used No � � Similarly Na�b � gives

M� ��

p� q�

a� b

p � q�

� � q�p

�� �q�pa�b�

Thus

Ni �

���������

a� bp � q �

�� �q�pi

� � �q�pa�b �i

p� q � p �� q

i�a� b� i� p � q

�

��� De�ne the eventsA�� r successes in n Bernoulli trials�B��success at the ith Bernoulli trial�C��r�� successes in the remaining n�� Bernoulli trials excluding

the ith trial�

P �A ��nr

�pr qn�r

P �B � p

P �C ��n� �r � �

�pr�� qn�r

We need

P �BjA �P �AB

P �A�

P �BC

P �A�

P �BP �C

P �A�

r

n�

��� There are�����

�ways of selecting �� cards out of �� cards� The

number of ways to select �� cards of any suit �out of �� cards equals�����

�� �� Four such �mutually exclusive suits give the total number

of favorable outcomes to be �� Thus the desired probability is given by

����

��

� � ���� � ���

�

��� �a Let n represent the number of wins required in � games so thatthe net gain or loss does not exceed ��� This gives the net gain to be

�� � n� � � n� � �

�� � n � ����

n � ��

P �net gain does not exceed �� ��� ��

� ���

��� ���

���� ����

P �net gain or loss exceeds �� � � � ���� � ����

�b Let n represent the number of wins required so that the net gainor loss does not exceed ��� This gives

�� � n��� � n

� � �

���� � n � �

P �net gain does not exceed �� �X��

n � ��

�� n

� ���

�n ���

����n

� ����

P �net gain or loss exceeds �� � �� ���� � ����

�

Problem Solutions for Chapter �

��� �a P �A occurs atleast twice in n trials

� � � P �A never occurs in n trials� P �A occurs once in n trials

� � � �� � pn � np�� � pn��

�b P �A occurs atleast thrice in n trials

� � � P �A never occurs in n trials� P �A occurs once in n trials

�P �A occurs twice in n trials

� � � �� � pn � np�� � pn�� �n�n� �

� p��� � pn��

���

P �doublesix ��

��

�

��

�

��

P ��double six atleast three times in n trials��

� � ���

� ����

�� �����

������ �

� ����

� �����

���� �

� ����

�� �����

��� ����

��� �a

p� � � ��

�

�� ����

�b

� ��

�

���

���

�

� �

�

�

�

��� ����

�c

���

�

���

���

�

� �

�

�

�

���

���

�

� �

�

� ��

��� ����