@Markiyan Hirnyk 

Yes, P[A] and P[B] are probabilities and take values between 0 and 1. I have this condition only P[A]<P[B].

H[A] >H[B] and they take positivie integer values between 0 to 50.

P>wu>w>c and they can take any values. Basically these are market price, wholesale prices(w and wu), and cost c. I prefer to keep them constant. Like p=1, wu=0.8, w=05 and c=0.1, and run the code on matlab and maple to see if I can say something about this inequality. This inequality is the comparison of two profit functions in 2 different situations.

@Axel Vogt 

Thank you Alex. I will go as much as I can, and include my intuitions.

@Carl Love 

Origial research. I can do it up to a certain point and stuck.

@Carl Love 

Thank you Carl. I ran the similar code on Matlab and got the same results. I need to prove it with another method. I mean instead of numerical analysis, I need to show matmematical terms with adding some assumptions maybe.

@Carl Love 

Hi Carl. Yes, by better I mean one profit function is higher for all the values.

Here are the constant and variables:

c=0.1, w=0.5, p=1,

Pb= 0:0.025:0.25 (Probability values and can take values from 0 up to 0.25 by 0.025 increments)

Pa = 0:0.025: (Hb - 0.25) (Probability values and can take values from 0 up to 0.25 by 0.025 increments but always less than Pb)

Hb:0:5:50  (These are quantities and can take values from 0 up to 50 by 5 increments)

Ha=0:5:(Hb-5)  (These are quantities and can take values from 0 up to 50 by 5 increments, but always less than Hb)

wu=0.5:0.1:1 (Take values from 0.5 up to 1 by 0.1 increments)

Pa<Pb and Ha<Hb

p≥wu>w≥c

These ranges and my assumptions.

Thank you so much in advance.

@Carl Love 

Hi Carl. I want to show that one function is better than the other. I am new to maple. I dont know coding but I wrote the functions and trying to take derivatives and plot them. It seem it is harder than I imagine.

Here are my functions. I plot them on matlab in a for loop and Profit B gave me better results. I want to prove it mathematically.

Profit*B = [Pb*(p-w)/(1-Pb)-c]*Hb+w*(Pb*(1-Pa)*Hb+Pa*Pb*Hb)+wu*Pa*(1-Pb)*Ha

Profit*C = [Pa*(wu-w+Pb*(p-wu))/(1-Pa)-c]*(Ha+Hb)+w*[Pa*(1-Pb)*Ha+Pb*(1-Pa)*Hb+Pa*Pb*(Ha+Hb)]

I assume that p,w, and c are contants. Pa, Pb, Ha, Hb, and wu are variables.