Dayaan M. answered • 3d
Earned A’s Twice in Precalculus | 5 Years of Tutoring Experience
In order to work out an expected value problem, it helps to first decide whose point of view you are taking, because the same situation looks different from each side. Here it is the company selling the warranty, so we want their expected gain to come out to zero.
Let p be the price they charge for the warranty.
Now think about the two things that can happen from the company's side.
If the product does not fail, which happens 99 percent of the time, they keep the price and pay out nothing. Their gain is p.
If the product does fail, which happens 1 percent of the time, they still keep the price but they have to hand over a $500 replacement. Their gain is p - 500.
Expected value means we multiply each outcome by how likely it is and add them up:
E = 0.99(p) + 0.01(p - 500)
Distributing the 0.01 gives:
E = 0.99p + 0.01p - 5
If you notice, 0.99p and 0.01p are like terms, and they add up to a full 1p, which makes sense because the company collects the price no matter what happens:
E = p - 5
Remember, our goal is to break even, so we set the expected value equal to zero:
p - 5 = 0
p = 5
There is a faster way to see this once the idea clicks. The expected payout per warranty sold is just 1 percent of $500, which is $5, so charging exactly $5 makes what comes in match what goes out on average.
So, our final answer is that they should charge $5. In real life a company would charge more than that, since breaking even leaves them nothing for administrative costs or profit, but $5 is the break even point the question is asking for.
