Best Buy offers a protection plan for new smartphones at 195

Best Buy offers a protection plan for new smartphones at $195. The absolutely most expensive iPhone you can buy right now is $849. Assume for a moment you are a very cautious but forgetful person: you would never, ever drop or damage your phone, but you might lose it. How likely must you be to lose your phone for $195 to be an actuarially fair price for mobile phone insurance?

Solution

Let probability of losing the phone be eaqual to \"p\". Thus, probability of not losing the phone is (1-p).

Now,

Expected Utility without Insurance = p(0) + (1-p) 849 ...............(1)

Expected Utility with Insurance = 849p + (1-p) 849 - 195...........(2)

(1) has to be equal to (2) for 195 to be the actuarially fair price for insurance.

Hence,

p(0) + (1-p) 849 = 849p + (1-p) 849 - 195

or, 849p = 195

or, p = 195/849 = 0.2297