Suppose that you are considering to start a business. You can invest your wealth of $100$ for it$.$ At $t($today$),$ you will buy assets needed for the business using the initial investment. You will run the business only for one period. In other words, the business ends at $t+1.$ You observe the business outcome at $t+1.$ The assets purchased at $t$ have zero remaining value at $t+1.$ You are the single owner of the company.

You expect the business will generate a profit of double of the initial investment in a good economic condition. You expect it will generate a profit of half of the initial investment in a bad economic condition. Ignore taxes.

$($a$)$ What is the profit and net gain in each scenario?