In the financial world, there are many types of complex instruments called derivatives that derive their value from the value of an underlying asset.
Consider the following simple derivative. A stock's current price is $80 per share. You purchase a derivative whose value to you becomes known a month from now. Specifically, let P be the price of the stock in a month.
If P is between $75 and $85, the derivative is worth nothing to you. If P is less than $75, the derivative results in a loss of 100*(75-P) dollars to you. (The factor of 100 is because many derivatives involve 100 shares.) If P is greater than $85, the derivative results in a gain of 100*(P-85) dollars to you.
Assume that the distribution of the change in the stock price from now to a month from now is normally distributed with mean $1 and standard deviation $8. Let P (big loss) be the probability that you lose at least $1000 (that is, the price falls below $65), and let P (big gain) be the probability that you gain at least $1000 (that is, the price rises above $95). Find these two probabilities. How do they compare to one another?