Expected Value of Event 
A valuable treasure worth 100,000 is lost in a large swamp and it must be removed in one day. I can hire one or more helicopters. Helicopters cost 1000 per day and have a 90% chance of finding my treasure. How many helicopters should I hire?

Here's my solution.
Probability of not finding the treasure is 10%. If I hire $n$ helicopters, probability of not finding the treasure is
$$\frac{1}{10^n}$$
Probability of finding the treasure is
$$1-\frac{1}{10^n}$$
If I find the treasure, I gain 100,000 and lose $1000 \times n$ on helicopters. Net gain is $100,000 - 1000n$.
If I don't find the treasure, net gain is $-100,000-1000n$.
Using the expected probability's definition, expected value after hiring $n$ helicopters is
$$\left(100,000-1000n\right)\left(1-\frac{1}{10^n}\right) + \left(-100,000-1000n\right)\left(\frac{1}{10^n}\right)$$
Using this expression, expected value for 2 helicopters is 96000 and expected value for 3 helicopters is 96800. I also graphed it and I found out that 3 helicopters has the highest expected value.
But my textbook says that the answer is 2. Please help me in finding my mistake.
 A: It is simpler to compute it as $(100,000)(1-\frac1{10^n}) - 1000n$,
since you have to shell out the money for the helicopters anyway.
For $n=2, E(X) = 97,000$
for $n = 3, E(X) = 96,900$
A: Because it's an initial condition that the treasure is lost, for the purposes of this problem, finding the treasure will result in a gain of 100,000 minus expenses and not finding the treasure will result in a loss equal to your expenses (not 100,000 + expenses). 
Your initial state is 100,000 lost, you have to minimize loss here. 
If you hire n helicopters (assumed independent), probability of finding the the treasure is (1 - Probability no helicopter finds the treasure) 
$ = (1 - \frac{1}{10^n}) $
So let's find our expected value:
$ E(\text{money gain}) = \text{Initial loss} + (100,000 - 1000n)*(1 - \frac{1}{10^n}) - (1000n * \frac{1}{10^n} )$
$= -100,000 + 10^5 - 10^{5-n} - n*10^3 + n*10^{3-n} - n*10^{3-n}  $
$=  - 10^{5-n} -n *10^3 $
If you take the global maximum of this, it's somewhere around 2.3
We can't send out 2.3 helicopters, so the answer has to be 2 or 3.
Value of expected gain at n = 2 is -1000 - 2000 = -$3000
At n = 3 it's - 100 - 3000 = -$3100
As a sanity check, n = 1 results in lower gain, and for anything higher than 3 the helicopters cost more than 3K so we can't beat -$3000.
Which is why n = 2 is the answer.
