Question 1196413
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A store specializing in mountain bikes is to open in one of two malls. 
If the first mall is selected, the store anticipates a yearly profit 
of $1,500, 000 if successful and a yearly loss of $500,000 otherwise. 
The probability of success is g. 

If the second mall is selected, it is estimated that the yearly profit 
will be $1.000,000 if successful, otherwise, the annual loss will be $300,000. 
The probability of success at the second mall is g. 

Which mall should be chosen in order to maximize the expected profit?
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<pre>
To facilitate my writing and your reading, I will operate 
with the thousand dollars as my monetary unit. It allows me forget about 
three zeroes at the end of numbers.


Math expectation of yearly profit for 1st mall is  1500g - 500(1-g) units,
which is  2000g - 500 units.


Math expectation of yearly profit for 2nd mall is  1000g - 300(1-g) units,
which is  1300g - 300 units.


So, we need compare these two quantities/expressions

    2000g - 500  ?  1300g - 300.


Here the sign  " ? "  stands for one of two possible inequality signs  " >= "  or  " <= ".

We transform it to

    2000g - 1300g  ?  500 - 300

         700g      ?     200

            g      ?     {{{200/700}}} = {{{2/7}}}.


Thus if g < 2/7, then 2nd mall gives greater Math expectation.

On contrary, if g > 2/7, then 1st mall gives greater Math expectation.

If g = 2/7, then Math expectations are the same for both malls.
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Solved.