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### Course: Precalculus > Unit 8

Lesson 9: Expected value- Mean (expected value) of a discrete random variable
- Mean (expected value) of a discrete random variable
- Interpreting expected value
- Interpret expected value
- Expected payoff example: lottery ticket
- Expected payoff example: protection plan
- Find expected payoffs
- Probability and combinatorics: FAQ

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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?(6 votes)
- Yes, this is the first question!

(This is the first answer!)(2 votes)

- 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.

2. Buy the protection plan

:)(3 votes)

## Video transcript

- [Instructor] We're told
that an electronic store gives customers the option of
purchasing a protection plan when customers buy a new television. That's actually quite common. The customer pays $80 for the plan and, if their television is
damaged or stops working, the store will replace it
for no additional charge. The store knows that 2% of
customers who buy this plan end up needing a replacement that costs the store $1,200 each. Here's a table that summarizes
the possible outcomes from the store's perspective. Let X represent the store's net gain from one of these plans. Calculate the expected net gain. So pause this video, see if
you can have a go at that before we work through this together. So we have the two scenarios here. The first scenario is
that the store does need to replace the TV
because something happens and so it's gonna cost
$1,200 to the store. But remember they got $80
for the protection plan. So you have a net gain of negative $1,120 from the store's perspective. There's the other scenario, which is more favorable for the store, which is a customer does
not need a replacement TV, so that has no cost and so their net gain is just the $80 for the plan. So to figure out the expected net gain, we just have to figure
out the probabilities of each of these and take
the weighted average of them. So what's the probability
that they will have to replace the TV? Well, we know 2% of
customers who buy this plan end up needing a replacement. So we could say this is two over 100 or maybe I'll write it as 0.02. This is the probability of X. And then the probability of not
needing a replacement, 0.98. And so our expected net gain is going to be equal to the probability of needing replacement times
the net gain of a replacement. So it's going to be times -$1,120. And then we're gonna
have plus the probability of not needing replacement, which is 0.98 times the net gain there. So that is $80. So we have 0.02x-1,120 is equal to that. And to that, we're gonna add, I'll open parentheses,
0.98x80, close parentheses, is going to be equal to $56. So this is equal to $56. And now you understand why the stores like to sell these replacement plans.