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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.

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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.