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# Expected payoff example: protection plan

We can find the expected payoff (or the expected net gain) of a protection plan offered by a store by taking the weighted average of the outcomes. Created by Sal Khan.

## Want to join the conversation?

• This is the first question?
• Yes, this is the first question!

• I think it is worth noting that it can still be worth getting a protection plan as a customer: chances are you won't need it, but if you don't buy one and do end up breaking it, \$1200 is a lot of money. Just because it's worth selling as a store doesn't nessicarily mean its not worth buying as a customer
(1 vote)
• You are correct. Just because it works out for the company doesn't mean it is not advantageous for the customer. We can actually use the reverse of this math to find how much money would be saved by each person. If 2% of customers who use this plane actually end up needing it, we get this payoff formula: -\$80 + (0.02)(\$1200).
We multiply the chance that the person needs the new TV (0.02) by the value the person would get from it (\$1200).
This gives us...
-\$80 + \$24 = -\$56
So, averaged out, if you buy the protection plan, you end up paying \$56 for it.
If we took this one step further and tried to analyze what percentage of people don't buy the protection plan but end up needing it? We could look at it this way: how likely would you need to be to actually break the TV to make the protection plan worth it?
We set up the equation, but this time, we will solve for the percent (x).
-\$56 = -\$1200(x)
the reason we use -\$56 is because that is the average amount a person buying the TV protection plan will lose. -\$1200 is the amount of money you will spend buying a new TV if your current one breaks. Isolating x by itself, we get this.
-\$56/(-\$1200) = x

Simplify:
56/1200 = x
x = 0.04667
Convert to Percent:
x = 4.667%

This means that if you have a 4.6667 or higher percentage of breaking your TV, it will be advantageous for you to buy the Protection plan. The original percentage is 2% of people will need the plan when they buy it. If we assume that those individuals who are actually going to need it (higher percent breakage) are the ones who buy the package, this population only has 2%. So, if you more than double the likelihood of breaking your TV than those who think they might actually break one, I have two things to say:
1. Shame on you.