Expected payoff example: protection plan | Probability & combinatorics | Khan Academy

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 two percent of customers who buy this plan end up needing a replacement that costs the store twelve hundred dollars each.

Here is 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 going to cost twelve hundred dollars to the store. But remember, they got eighty dollars for the protection plan, so you have a net gain of negative one thousand one hundred and twenty dollars from the store's perspective.

There's the other scenario, which is more favorable for the store, which is the customer does not need a replacement TV. So that has no cost, and so their net gain is just the eighty dollars for the plan.

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 two percent of customers who buy this plan end up needing a replacement. So we could say this is 2 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 is 0.98.

And so, our expected net gain is going to be equal to the probability of needing a replacement times the net gain of a replacement. So it's going to be times negative one thousand one hundred and twenty dollars, and then we're going to have plus the probability of not needing a replacement, which is 0.98 times the net gain there, so that is $80.

So we have 0.02 times negative one thousand one hundred and twenty, and that we're going to add. I'll open parentheses, 0.98 times eighty, closed 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.