Random vaiables 2

A and B play the following game:A writes down either number 1 or number 2 ,and B must guess which one.If the number that A has written down is i and B has guessed correctly,B receives i units from A.If B makes a wrong guess,B pays 3/4 unit to A.If B randomizes his decision by guessing 1 with probability p and 2 with probability 1-p,determine his expected gain if
(a)A has written down number 1
Expected Gain= p(1)+(1-p)(-3/4)
=p-3/4+3/4p
=7/4 p -3/4

(b) A has written down number 2
Expected Gain =p(-3/4)+(1-p)(2)
=-11/4 * p +2

What value of p maximizes the minimun possible value of B's expected gain and what is this maximum value?

Assume the probability of A writes down 1 is q and writes down 2 is 1-q, i.e. P(1)=q; P(2)=(1-q)
Then expected gain (E)
=q* (7/4 * p -3/4) + (1-q) (-11/4*p+2)
=7/4* pq-3/4*q-11/4 *p+2+11/4*pq-2q
=18/4*pq -11/4*q-11/4 * p+2
=(18/4*q-11/4)p-11/4q+2
=(9/2*q-11/4)p-11/4*q+2

It depends on the value of 9/2 *q -11/4
If 9/2* q -11/4>0 =>q>11/18, then set p=1
If 9/2*q-11/4<0 => q<11/18, then set p=0

If q=0.5, then set p=0,
max gain =-11/4 (0.5) + 2
=-11/8+2
=5/2 units