Should one pay buy a warrenty? (What is the conclusion?) This problem is from khan academy:


  John just bought a brand new cell phone and is considering buying a
  warranty. The warranty costs 200 euros and is worth 1000 euros if his
  phone breaks. John estimates that there is a 10%, percent chance of
  his phone breaking.
  
  Find the expected value of buying the warranty.


Step by step guide to the solution



Now, the reason why I created this question is because I am unsure of what the conclusion is. I assume that since the expected value is -100 euros, it means that we can expect to lose 100 euros by buying this warrenty; thus, we should not buy the warrenty. 


Is my assumption (conclusion) correct?




expected value function, where x = the probability of braking phone

Intersection with the x axis


Would it be correct to assume that:


if the probability of the phone braking is over 20%, we should buy a warrenty, and simiarily avoid buying one if the probability is less than 20%.
I'm not sure if you are asking this in a mathematical way about the "conclusion" (I would say the only mathematical conclusion is that the company can expect to make an average of \$100 on each warranty sold, although they probably make more; to really know this you would need to know how much it costs them to replace John's phone), or about the practical lesson to take regarding John.

For John, you need to consider what happens if you don't have the warranty. In other words, you should compare with the expected loss of buying the phone without the warranty.

But even that isn't the whole story. Of course the expected value of buying a warranty will be less for the consumer than that of not buying the warranty; that's how companies make money selling them. The real issue is whether John can afford to lose \$1000. If you were to lose \$1000 today, how would that affect your life? How would spending \$200 today affect your life? Can John afford to take a 10% chance of losing \$1000? If John is wealthy enough to take these kinds of losses in stride, he is better off not buying warranties. Otherwise it makes sense to spend an amount he can afford to prevent a loss he cannot afford.
The expected overall value of the warranty, as calculated, is -100 Euros.

On the other hand, assuming that without the warranty John will still purchase a new phone if his current one breaks, then there is a 90% chance John will pay nothing and a 10% chance he will have to pay 1000 Euros, making the expected value of not buying the warranty also -100 Euros.

In this case, as it turns out, there is no change in the value between buying and not buying the warranty so based on these calculations it doesn't matter if he does or not. However, if John valued certain outcomes differently (e.g. he doesn't want to have to pay large chunks of money, or he hates filling out forms), then you would need to used a different measure of the expected utility of the two options.