Verizon offers a protection plan for new smartphones at 195

Verizon 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? [Hint: you’re solving for p, the probability of losing your phone. Not losing your phone, therefore, happens 1-p percent of the time.]

Solution

Suppose that you lose the phone. In that case, the loss for verizon = 849 - 195 = $654

So verizon will hope that you do not lose your phone in 654/849 or 77.03% of times

So probability of losing your phone = 1 - 77.03% = 22.97%